Bandwidth adjustable euclidean modulation

ABSTRACT

A method and apparatus for using Euclidean modulation in an antenna are disclosed. In one embodiment, a method for controlling an antenna comprises mapping a desired modulation to achievable modulation states, mapping modulation values associated with the achievable modulation states to one or more control parameters, and controlling radio frequency (RF) radiating antenna elements using the one or more control parameters to perform beam forming.

PRIORITY

The present patent application claims priority to and incorporates byreference the corresponding provisional patent applications No.62/940,423, titled, “BANDWIDTH ADJUSTABLE EUCLIDEAN MODULATION,” filedon Nov. 26, 2019.

FIELD OF THE INVENTION

Embodiments of the present invention relate to the field of antennas;more particularly, embodiments of the present invention relate to theuse of a distance-based mapping between desired and achievablemodulation states to control a holographic beamforming antenna.

BACKGROUND

Metasurface antennas have recently emerged as a new technology forgenerating steered, directive beams from a lightweight, low-cost, andplanar physical platform. Such metasurface antennas have been recentlyused for microwave applications and stand to provide relevant hardwareand software gains for various applications such as microwave imaging,communications and synthetic aperture radar.

Metasurface antennas may comprise a waveguide structure, loaded withresonant, complementary metamaterial elements in the upper surface ofthe waveguide that can selectively couple energy away from a guided waveinto free space as radiation. By tuning the constituent elements'characteristics, a hologram at the aperture plane can be achieved, inwhich the waveguide mode acts as the reference wave and the collectionof tuned elements form the hologram. The overall radiation from theseholographic antennas can thereby be modulated to form arbitrary patternsby the use of electronic tuning. These antennas are capable of achievingcomparable performance to phased array antennas from an inexpensive andeasy-to-manufacture hardware platform.

By using simpler elements as compared to phased arrays, the operation ofmetasurfaces is easier and faster. These elements, however, do notexhibit the same level of control as is achievable with phase shiftersand amplifiers, common to phased array architectures. To regain some ofthe control possible with phased array elements, metasurface elementsare typically spaced closer together in order to more finely sample theguided wave. In further contrast to phased array antennas, however,tuning metamaterial elements does not provide independent control ofboth the magnitude and phase of each individual element in the array.Instead, tuning a metamaterial unit cell results in a shift in theresonant frequency, which shifts both the magnitude and phase responsewith only one control knob. As a result, attempting to create arbitrarymagnitude or phase patterns within a metasurface antenna can yieldundesirable results.

The reason that creating arbitrary magnitude or phase patterns does notwork as intended can be traced back to the coupled nature of themagnitude and phase of the resonant elements in a metasurface antenna.Considering a phase pattern, when tuning an antenna element to a certainphase value, the magnitude of the element becomes correspondinglyshifted. This arbitrary shift can lead to unwanted periodic behavior orto low radiation efficiency. In the same manner, similar side effects ofphase artifacts result when attempting to create a magnitude pattern.These problems do not exist in traditional phased array systems becauseamplifiers and phase shifters can directly compensate for any suchunwanted artifacts.

One method of modulating individual elements in a metasurface antennaelement is to turn some elements “on” while others are kept “off”. Thisis referred to as binary modulation. Another method that is used isreferred to as greyshade modulation in which antenna elements have morestates than simply one “on” states and an “off” state.

Instantaneous bandwidth (IBW) is defined herein as the frequency rangethat the antenna demonstrates consistent radiation characteristicswithout changes in the modulation pattern, where consistency refers togain variation being below a defined requirement in response to changesin the modulation pattern (e.g., changes in frequency). In holographicmetasurface antennas, the modulation pattern refers to as the impedanceof the scattering antenna elements (RF radiating antenna elements)forming the metasurface. The desired impedance for each element iscalculated using a holographic equation that relates the feed wave to adesired radiation pattern. These impedance values, however, are idealand need to be translated to an impedance that can be realized using theavailable scattering antenna elements. This is done using Euclideanmapping that reduces, and potentially minimizes, the Euclidean distance(i.e., mapping error) between the desired and achievable impedances.

One way to do this calculation to reduce the Euclidean distance is at asingle desired frequency, which can be the center frequency, themodulation frequency, or the carrier frequency. This means that thecalculated modulation pattern is optimized only for that frequency andnot necessarily optimized at any other frequency. Because of this, asthe frequency moves further away from that desired frequency, mappingerrors can result in an offset in the direction of the beam compared tothe beam at the reference frequency. This is generally referred to asbeam squinting. Beam squinting reduces the gain flatness, whichconsequently limits the instantaneous bandwidth.

SUMMARY

A method and apparatus for using Euclidean modulation in an antenna aredisclosed. In one embodiment, a method for controlling an antenna havinga plurality of radio frequency (RF) radiating antenna elements,comprises: performing modulation for a frequency range, includingdetermining a first set of modulation values for achievable modulationstates associated with the frequency range; mapping modulation valuesassociated with the achievable modulation states to one or more controlparameters; and controlling the radiating antenna elements using the oneor more control parameters to perform beam forming.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be understood more fully from the detaileddescription given below and from the accompanying drawings of variousembodiments of the invention, which, however, should not be taken tolimit the invention to the specific embodiments, but are for explanationand understanding only.

FIG. 1 illustrates desired polarizabilities required to achieve adirective beam and polarizabilities achievable by tuning metasurfaceelements.

FIG. 2 illustrates Lorentzian polarizability for the idealizedmetamaterial element used to model a sample metasurface antenna in threedifferent tuning states.

FIGS. 3A-C illustrate the a) phase hologram and b) Euclidean modulation.

FIG. 4 illustrates far-field radiation patterns for a phase hologram-and Euclidean modulation-optimized broadside beams across theoperational bandwidth.

FIG. 5 illustrates far-field radiation patterns for a phase hologram-and Euclidean modulation-optimized beams steered to 25 degrees acrossthe operational bandwidth.

FIG. 6 illustrates beam characteristics for three modulation techniquesover a wide bandwidth.

FIG. 7 illustrates beam characteristics for three modulation techniquesover a wide bandwidth.

FIG. 8 illustrates an example of the Euclidean modulation pattern inwhich the ideal/required polarizabilities are approximated by theLorentzian curve by mapping the desired polarizabilities to the nearestpossible points on the curve possible polarizabilities.

FIGS. 9A-D illustrate an algorithm for determining optimized resonatorstates using the Euclidean modulation principle.

FIG. 10A illustrates binary modulation far-field pattern in the ϕ=0plane.

FIG. 10B illustrates greyshade modulation far-field pattern in the ϕ=0plane.

FIG. 10C illustrates Euclidean modulation far-field pattern in the ϕ=0plane.

FIG. 10D illustrates Euclidean Modulation Element Density

FIG. 11A illustrates return loss and end-of-guide loss .

FIG. 11B illustrates maximum directivity over frequency.

FIG. 11C illustrates a histogram of resonators states for a greyshademodulation scheme.

FIG. 11D illustrates return loss and end-of-guide loss.

FIG. 11E illustrates maximum directivity over frequency.

FIG. 12 illustrates a calculation of a required average polarizabilitydown the length of a waveguide in order to obtain an even aperture.

FIGS. 13A-D illustate how Euclidean modulation is altered byrestriction.

FIG. 14 illustrates calculation of the imaginary part of the averagepolarizability <Im{α}> as a function of the maximum allowed imaginarypolarizability α _(max) for a restricted euclidean modulation patternapplied to metamaterial elements with A=0.2 mm³.

FIG. 15 illustrates a plot of the required restriction, α _(max) (curve2301), which is required to obtain a particular average imaginarypolarizability, <Im{α}> (curve 2302), that will result in an evenaperture.

FIG. 16 illustrates far-field pattern of a waveguide with an effectiveindex of n_(wg)=1.15, under both restricted and unrestricted Euclideanmodulation, when the maximum directivity of the restricted Euclideanbeam is 21.2 dBi, and the maximum directivity of the unrestrictedEuclidean beam is 20.2 dBi.

FIG. 17 shows the analysis applied to the same antenna, but modified sothat n_(wg)=1.4.

FIGS. 18A-C illustrate (a) the feed wave decay, (b) the magnitude of thepolarizability of each metamaterial element, and (c) the dipole momentof each metamaterial element in the waveguide under restricted andunrestricted Euclidean modulation.

FIG. 19 illustrates traditional Euclidean mapping at 10.4 GHz for arandom element over the aperture.

FIG. 20 illustrates bandwidth adjustable Euclidean mapping compared withtraditional Euclidean mapping.

FIG. 21 illustrates a comparison of gain achieved for traditionalEuclidean modulation at 10.4 GHz and gain achieved for bandwidthadjustable Euclidean modulation with an example calculation window.

FIG. 22A is a flow diagram of one embodiment of a process for performingbandwidth adjustable Euclidean modulation processing.

FIG. 22B is a flow diagram of another embodiment of a process forperforming bandwidth adjustable Euclidean modulation processing.

FIG. 22C is a flow diagram of yet another embodiment of a process forperforming bandwidth adjustable Euclidean modulation processing.

FIG. 23 illustrates an aperture having one or more arrays of antennaelements placed in concentric rings around an input feed of thecylindrically fed antenna.

FIG. 24 illustrates a perspective view of one row of antenna elementsthat includes a ground plane and a reconfigurable resonator layer.

FIG. 25A illustrates one embodiment of a tunable resonator/slot.

FIG. 25B illustrates a cross section view of one embodiment of aphysical antenna aperture.

FIGS. 26A-D illustrate one embodiment of the different layers forcreating the slotted array.

FIG. 27 illustrates a side view of one embodiment of a cylindrically fedantenna structure.

FIG. 28 illustrates another embodiment of the antenna system with anoutgoing wave.

FIG. 29 illustrates one embodiment of the placement of matrix drivecircuitry with respect to antenna elements.

FIG. 30 illustrates one embodiment of a TFT package.

FIG. 31 is a block diagram of an embodiment of a communication systemhaving simultaneous transmit and receive paths.

FIG. 32 is a block diagram of another embodiment of a communicationsystem having simultaneous transmit and receive paths.

DETAILED DESCRIPTION

In the following description, numerous details are set forth to providea more thorough explanation of the present invention. It will beapparent, however, to one skilled in the art, that the present inventionmay be practiced without these specific details. In other instances,well-known structures and devices are shown in block diagram form,rather than in detail, in order to avoid obscuring the presentinvention.

To overcome problems disclosed above, an antenna is controlled usingEuclidean Modulation. Euclidean modulation is a technique that canimprove and potentially optimize the tuned resonant frequency of eachmetamaterial element to achieve arbitrary, desired, radiation patternsfrom a metasurface antenna. More specifically, Euclidean modulation is aparticular technique for finding a resonance frequency for eachmetamaterial element that provides a superior overall beam performancecompared to prior art. While metasurface antennas do not exhibit thesame direct control over the element responses, employing theappropriate modulation technique results in performance on par withphased arrays that can be achieved from a hardware platform that islightweight, low power, and inexpensive.

In one embodiment, the Euclidean modulation is a bandwidth adjustableEuclidean modulation. In one embodiment, the bandwidth adjustableEuclidean modulation selects for use a Euclidean modulation that meetsIBW requirements. In another embodiment, the bandwidth adjustableEuclidean modulation selects for use a Euclidean modulation that meetsboth gain and IBW requirements. One or both of the gain and IBWrequirements may be set by a satellite services provider.

A method and apparatus for performing bandwidth adjustable Euclideanmodulation with satellite antennas in a satellite communication systemare disclosed. In one embodiment, the satellite antennas areelectronically-steered flat panel antennas with radio frequency (RF)radiating elements, such as, for example, but not limited to, theantennas described in more detail below.

The following description provides a general description of holographicantennas, phase hologram modulation, followed by a description ofEuclidean modulation, and then a description of bandwidth adjustableEuclidean modulation.

Holographic Antennas

In one embodiment, holographic antennas comprise a waveguide structurewith a guided wave incident on the metasurface elements embedded withinthe waveguide. Relative to a traditional hologram, the guided wave canbe treated as the reference wave, which interferes with the array ofmetasurface elements. By tuning these elements, the hologram can bealtered to produce arbitrary radiation patterns in the far field.

The scattered field of any object or antenna can be described by theobject's far-field pattern. In the case of a leaky waveguide antenna orholographic meta-material antenna, the antenna performs ideally when itradiates in only a single direction, i.e., when the field produced bythe antenna is a plane wave, so that

E(r)=E ₀ e ^(−ik) ⁰ ^(−r).   (1)

where E₀ is the polarization and ko is the desired direction of theantenna beam. In the language of scattering theory, this would implythat the radiation pattern from the antenna is a delta function ink-space. If the antenna were to extend infinitely in a plane S withunit-normal vector {circumflex over (n)}, then the set of currents thatwould produce this far-field pattern are a set of magnetic surfacecurrents over a perfectly electric conducting (PEC) ground plane, givenfor points r ∈ S as

K _(M) =E×{circumflex over (n)}=E ₀ ×{circumflex over (n)}e ^(−ik) ⁰ −r.  (2)

Alternatively, one way to prove equation (2) is to use Schelkunoff sequivalence principles. For more infomration, see R. E. Collin,Antennas, and P. Society, Field theory of guided waves, IEEE/OUP serieson electromagnetic wave theory, New York: IEEE Press, 1991, publishedunder the sponsorship of the IEEE Antennas and Propagation Society. Oneof Schelkunoffs equivalence principles states that, if the electric andmagnetic fields are known on the boundary S of a domain V that containsno sources, then the field inside V is equal to the field that would beradiated by a magnetic current K_(M)=E×n that lies on S, and is backedby a PEC just behind S. However, it is also possible to stateSchelkunoffs equivalence principle backwards: if one desires aparticular electric field E in a volume V that is bounded by a surfaceS, then placing a magnetic surface current of K_(M)=E×n along with a PECwill achieve the electric field E, provided that E is a solution to thewave equation in free space. If the volume V to be the upper half-spacewith S being the plane where z=0 , then the inverse of Schelkunoff sequivalence principle dictates that the magnetic current distribution inequation (2) will radiate like a plane wave in the direction k₀. Theinverse of Schelkunoffs equivalence principles is not unique: i.e.,there are other current distributions that will yield the same field(for instance, an electric surface current along with a perfect magneticconductor (PMC) boundary applied to the surface S will achieve the sameresult), but it does give a solution to the problem.

In the context of holographic metamaterial antennas, the goal is toproduce the required surface current K_(M) using waveguide slots orcomplimentary metamaterial elements. In one embodiment, these elementsare known to radiate like magnetic dipoles, and since they are coupledto a ground plane, they can provide the required magnetic surfacecurrent to produce an ideal plane-wave beam in the limit that theantenna is infinitely large. Holographic metamaterial antennas typicallyuse a waveguide or feed structure to excite the complimentarymetamaterial elements, and in the limit that the distance A between themetamaterial elements is deeply sub-wavelength, the magnetic surfacecurrent can be written down as a magnetic surface susceptibility tensorχ _(M), times the magnetic field of the feed wave H_(f):

K _(M)(r)=−iω{circumflex over (χ)} _(M)(r)H _(f)(r).   (3)

The surface susceptibility is defined as the magnetic dipole momentgenerated per unit area on the surface of the antenna, and so in oneembodiment it is physically composed of a lattice of complementarymetamaterial elements in the surface of the waveguide that scatter likemagnetic dipoles. In order to achieve a well-formed beam, the surfacecurrent is set equal to E₀×ne^(k) ⁰ ^(·r), which is done in practice bydesigning the surface susceptibility tensor {circumflex over (χ)}_(M)using the modulation pattern. Moreover, in order to allow the antenna toproduce a beam with any polarization, both eigenvectors of the surfacesusceptibility tensor are orthogonal and controlled in two differentdirections. In one embodment, this is done using holographicmetamaterial antennas by imposing two lattices of metamaterialscattering elements, where each element in each lattice can be excitedwith a magnetic dipole moment in only one direction, and the directionsof the dipoles in the two lattices are orthogonal.

To derive the control of these lattices of magnetic dipoles, considertwo lattices where the positions of the dipoles in the first lattice arer_(i) with orientations {circumflex over (v)}_(i), and where thepositions of the dipoles in the second lattice are also r_(i), but withorientations {circumflex over (μ)}_(i) such that {circumflex over(v)}_(i) ·{circumflex over (μ)} _(i)=0. Then the surface susceptibilitytensor can be generally written as

χ _(M)(r _(i))=χ₁(r _(i)){circumflex over (v)} _(i) ⊗{circumflex over(v)} _(i)+χ₂(r _(i)){circumflex over (μ)}_(i)⊗{circumflex over (μ)}_(i)  (4)

where the symbol ⊗ is used to designate the tensor product. If thescattering of the elements into the waveguide is small enough to benegligible, then the surface susceptibility at position r_(i) is relatedto the magnetic polarizability α_(i) ¹ of the element in the firstlattice at position r_(i) by α_(i) ¹=Λ²χ_(i)(r_(i)) while thepolarizability of the element in the second lattice is likewise α_(i)²=Λ²χ₂(r_(i)).Using this relationship, together with equations (3) and(4), a straightforward calculation shows that the requiredpolarizabilities of the dipoles are

$\begin{matrix}{\alpha_{1}^{i} = {\frac{i\; \mu_{0}\Lambda^{2}}{Z_{0}k}{E_{0} \cdot \left( {\hat{n} \times {\hat{v}}_{i}} \right)}\left( \frac{e^{{ik}_{0} \cdot r_{i}}}{H_{f} \cdot {\hat{v}}_{i}} \right)}} & \left( {5a} \right) \\{\alpha_{2}^{i} = {\frac{i\; \mu_{0}\Lambda^{2}}{Z_{0}k}{E_{0} \cdot \left( {\hat{n} \times {\hat{\mu}}_{i}} \right)}{\left( \frac{e^{{ik}_{0} \cdot r_{i}}}{H_{f} \cdot {\hat{\mu}}_{i}} \right).}}} & \left( {5b} \right)\end{matrix}$

At this point, a particular form of the feed wave needs to be chosen. Inone embodiment, the feed structure is a linear waveguide that only has asingle propagating mode, and then the feed-wave on the upper surface ofthe waveguide is proportional to H_(f)∝h_(f)e^(iβy), where β is thepropagation constant, and the waveguide is assumed to propagate waves inthe y-direction. In this case, the requirement on the polarizabilitiesto produce an ideal beam-pattern is

$\begin{matrix}{\alpha_{1}^{i} = {\frac{i\; \mu_{0}\Lambda^{2}}{Z_{0}k}\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{v}}_{i}} \right)}{h_{f} \cdot {\hat{v}}_{i}} \right)e^{{i{({k_{0} - {\beta \; y}})}} \cdot r_{i}}}} & \left( {6a} \right) \\{\alpha_{1}^{i} = {\frac{i\; \mu_{0}\Lambda^{2}}{Z_{0}k}\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{\mu}}_{i}} \right)}{h_{f} \cdot {\hat{\mu}}_{i}} \right){e^{{i{({k_{0} - {\beta \; y}})}} \cdot r_{i}}.}}} & \left( {6b} \right)\end{matrix}$

Notice that the term

$\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{v}}_{i}} \right)}{h_{f} \cdot {\hat{v}}_{i}} \right) \equiv Z_{ant}$

has units of impedance, and it defines the amplitude of the electricfield of the radiated wave relative to the amplitude of the magneticfield of the feed wave.

Also, while the modulation pattern is defined above by desired andachievable polarizabilities, it may be defined by the desired andachievable/realized/available impedances. As is shown, thepolarizabilities have a unit of impedance since it defines the amplitudeof electric field of the radiated wave to the amplitude of the magneticfield of the feed wave. Thus, it should be understood that thesedefinitions of the modulation pattern are interchangable.

As the index i runs through various metamaterial element positionsr_(i), the ideal modulation pattern is proportional to e^(i(k) ⁰^(−β)·r) ^(i) , which traces out a unit circle in the complex plane.Hence, in an ideal situation, all of the elements would be excited at anequal level, but with different scattering phases which are dictated bythe modulation pattern.

Alternatively, in one embodiment, the feed structure is a planarwaveguide that is centrally fed, and the feed wave follows the formH_(f)=h_(f){circumflex over (θ)}H⁽²⁾(βr)≈h_(f){circumflex over(θ)}e^(iβr)/√{square root over (βr)}.

$\begin{matrix}{\alpha_{1}^{i} = {\frac{i\; \mu_{0}\Lambda^{2}}{Z_{0}k}\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{v}}_{i}} \right)}{h_{f}{\hat{\theta} \cdot {\hat{v}}_{i}}} \right)\sqrt{\beta \; r_{i}}e^{{i({k_{0} - \beta}\;)} \cdot r_{i}}}} & \left( {7a} \right) \\{\alpha_{1}^{i} = {\frac{i\; \mu_{0}\Lambda^{2}}{Z_{0}k}\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{\mu}}_{i}} \right)}{h_{f}{\hat{\theta} \cdot {\hat{\mu}}_{i}}} \right)\sqrt{\beta \; r_{i}}{e^{{i({k_{0} - \beta}\;)} \cdot r_{i}}.}}} & \left( {7b} \right)\end{matrix}$

In this case the required polarizability rotates through all possiblephases, but increase as √{square root over (r)} to compensate for thenatural spread in the feed wave through the aperture. In both of thesecases, the requirement for an ideally radiating antenna is that allpossible phases of the polarizability of the scattering elements beavailable. Unfortunately, merely changing the resonance frequency of ametamaterial element does not only change the phase but also themagnitude of scattering of the element, and it is only possible toobtain 180 degrees of all the possible scattering phases at anymagnitude. To show this, consider the general form for thepolarizability of a resonant metamaterial element. A resonant magneticdipole over a ground plane has a polarizability that follows aLorentzian function, which is the form

$\begin{matrix}{\alpha = \frac{\mu_{0}\omega_{0}^{2}A}{\omega_{0}^{2} - \omega^{2} - {i\; {\omega \left( {\Omega + {\omega^{2}\omega_{0}^{2}{A/3}\pi \; c^{3}}} \right)}}}} & (8)\end{matrix}$

where ω₀=1/√{square root over (LC)} is the resonance frequency of themetamaterial element, c is the speed of light in free space, and Ω isthe Ohmic loss rate. The constant A is an arbitrary coupling coefficientthat has dimensions of volume and roughly corresponds to the cube of theeffective radius of the dipole. The term iω²ω₀ ²A/3πc² in thedenominator corresponds to the radiative loss rate, and here it ismodified to take into account the losses when the dipole is placed overa ground plane rather than in free space. For more information onLorentzian, see M. Albooyeh, D. Morits, and S. A. Tretyakov, “Effectiveelectric and magnetic properties of metasurfaces in transition fromcrystalline to amorphous state,” Phys. Rev. B, vol. 85, p. 205110, May2012.

This discussion now leads to the definition of the basic difficulty withholographic metamaterial antennas that Euclidean modulation attempts tosolve. The image of equation (8) as a function of the resonancefrequency ω₀ represents the range of polarizabilities that may beachieved by tuning the resonance frequency of a metamaterial cell.However, equations (6a), (6b), (7a) and (7b) prescribe thepolarizabilities that are required for each metamaterial element inorder for the antenna to radiate in a far-field pattern that mostclosely approximates an ideal plane wave. Unfortunately, the set ofachievable polarzabilities does not overlap with the set of requiredpolarizabilities. In FIG. 1, the real and imaginary parts of equation(8) are plotted as curve 101 in the complex plane that is parameterizedby ω₀ for the choice of parameters Ω=0 and A=(λ/10)³, and hence thisillustrates the range of achievable polarizabilites using metamaterialelements. The same plot also shows the required polarizabilities fromequations (6a) and (6b) for a linear waveguide with metamaterialelements that are placed Λ=λ/3.6 apart, and with Z_(ant)=3.1Z₀.

Unfortunately, only one out of the 18 polarizabilities needed for thismodulation pattern actually lies on the curve of achievablepolarizabilities. In order to achieve good beam performance, in oneembodiment, an approximation is made in order to use the set ofachievable polarizabilities to approximate the requiredpolarizabilities. The Euclidean Modulation technique described herein isused to choose points out of the set of achievable polarizabilities thatbest approximate the required polarizabilities.

Phase Hologram Modulation

The primary existing method for optimizing the polarizability of eachmetamaterial element is to form a phase hologram. In this approach, thedesired phase of the surface currents still follows a circle in thecomplex plane as given by equations (6a) and (6b). However, the state ofthe unit cell is chosen by taking the phase of eq. (8) and finding thepoint that agrees in phase with equations (6a) and (6b). In this way themagnitude of the unit cell's response is ignored, but the correct phaseis chosen. In the case of a metasurface antenna, using a phase hologramto create a beam results in some unintended consequences due to thecoupled phase and magnitude of the tunable radiating elements. The goalof achieving a desired phase profile may be met, but in doing so, manyelements with the correct phase will be incidentally set to have a lowmagnitude. Since metasurface antennas lack amplifiers to compensate forthis type of side effect, using a traditional phase hologram will resultin lost efficiency, which can further be magnified by the losses in aspecific system. Conversely, one might focus on a single phase inequations (6a) and (6b), by shifting the circle up the imaginary axis onthe complex plane and taking the imaginary part while ignoring the realpart of equations (6a) and (6b) to read

$\begin{matrix}{\alpha_{1}^{i} = {\frac{\mu_{0}\Lambda^{2}}{Z_{0}k}{\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{v}}_{i}} \right)}{h_{f} \cdot {\hat{v}}_{i}} \right)\left\lbrack {{\cos \left( {\left( {{\beta \hat{y}} - k_{0}} \right) \cdot r_{i}} \right)} + 1} \right\rbrack}}} & \left( {9a} \right) \\{\alpha_{1}^{i} = {\frac{\mu_{0}\Lambda^{2}}{Z_{0}k}{\left( \frac{E_{0} \cdot \left( {\hat{n} \times {\hat{\mu}}_{i}} \right)}{h_{f} \cdot {\hat{\mu}}_{i}} \right)\left\lbrack {{\cos \left( {\left( {{\beta \hat{y}} - k_{0}} \right) \cdot r_{i}} \right)} + 1} \right\rbrack}}} & \left( {9b} \right)\end{matrix}$

This somewhat artificial construct modifies the modulation circle ofdesired polarizabilities in the complex plane by mapping it onto a setof positive, real numbers. However, once this is done, it's possible toignore the phase of the desired modulation pattern, and hence selecttuning states of the elements by finding the amplitude of the set oftunable polarizabilities from the Lorentzian elements, which are givenby taking the magnitude of equation (8). These amplitudes are shown inFIG. 2. An amplitude hologram is therefore achieved by finding thepoints in the curve of tunable polarizabilities that are closest inamplitude to the amplitude prescribed in equations (9a) and (9b). Asopposed to the phase hologram, which only makes one approximation byignoring the magnitude, amplitude hologram therefore has two steps inapproximation. The first approximation in amplitude hologram is that itignores the phase of the response of the elements, and the secondapproximation in amplitude hologram is that it artificially maps theideal modulation pattern equation to the positive real axis, so thatonly the amplitude of the response of the elements is relevant.

Euclidean Modulation

Euclidean Modulation is one technique to approximate the set of requiredpolarizabilities with tunable elements in a metasurface. To do this, theEuclidean distance between the required and achievable polarizabilitiesis reduced, and potentially minimized. FIGS. 3A-3C illustrate a) phasehologram and b) Euclidean modulation. Referring to FIG. 3A-3C, the phasehologram approach involves minimizing the error between the phase of thedesired and available polarizabilities. FIG. 3A illustrates where eachpoint on the set of required polarizabilities is mapped to the nearestpoint on the Lorentzian curve. This mapping is defined by the resonancefrequency ω_(0i) for the i^(th) metamaterial element located at positionr_(i) such that the distance

d=|α _(L(ω) _(0i) ₎−α_(D)(r _(i))|  (10)

is a global minimum. FIGS. 3B and 3C illustrate the Euclidean modulationapproach, which involves minimizing the Euclidean norm between thedesired and available polarizabilities. Minimizing the Euclideandistance between those two points is a way of locally minimizing theerror between the required polarizability and the actual polarizabilitydelivered by the dipole. This does not guarantee that the modulationpattern chosen by Euclidean Modulation is the globally best possiblepattern, but it is an optimized pattern in the sense that it is a localmaximum of the directivity function.

Far-Field Performance

To examine the far-field performance of Euclidean Modulation as comparedto the phase hologram and magnitude hologram approaches, consider asampled, analytically modeled metasurface. In one embodiment, thismetasurface consists of a waveguide that is aligned along the x-axiswith 50 complementary metamaterial elements that are placed a distanceΛ=λ/4 apart from each other, where λ is the free space wavelength. Theseelements are modeled with an idealized Lorentzian polarizability, asplotted in FIG. 2. To model the antenna, an incident magnetic field ismodeled as H(x)=H₀e^(iβx). This guided wave is assumed to be unperturbedby the scattering of the elements in the waveguide (without includingthe aperture taper). By exciting each metamaterial element with theguided wave, a magnetic dipole moment is induced as a function of x andthe tuned polarizability, as determined by the given modulationtechnique. To find the resulting far-field radiation patterns, themagnetic dipole moments of the elements mi are calculated by assumingthe feed wave is unperturbed as it flows through the waveguide, andhence each dipole moment is given by m_(i)=α_(t)(ω_(0,i))H(x_(i)), whereω_(0,i) is the tuned resonance frequency of the i^(th) unit cell, asprescribed by the chosen modulation technique. The collection of dipolemoments can be propagated into the far field using Green's functions.Sample radiation patterns for phase hologram and Euclidean Modulationoptimization approaches are shown in FIG. 4 for different frequencies.The Euclidean Modulation patterns result in beams with superiorbroadband directivity and reduced sidelobe levels, as described infurther below.

Broadband Directivity

The beam patterns shown in FIG. 4 demonstrate that Euclidean modulationexhibits superior directivity across a wide bandwidth. The directivityof beams generated with a phase hologram and Euclidean Modulation can becalculated across the full operational bandwidth. The only intrinsiclimitation on the bandwidth of these patterns is the beam squint. If apattern is a solution to the modulation equations at one particularfrequency for a certain angle, then for a small change in frequency thatpattern will be a solution to the modulation pattern equations at aslightly different angle. This effect causes the beam to shift its angleas the frequency changes, which is consistent with the operation of aconventional leaky-wave antenna. That is, by creating the appropriatehologram at the surface of the aperture by tuning the metasurfaceelements, a leaky-wave antenna has essentially been created. Note thatthe tuning can be adjusted per frequency to maintain a constant steeredangle.

An additional limitation on the broadband performance of the antenna isdue to the decay of the fields in the waveguide. The ideal surfacecurrent on the antenna would follow equation (2), which prescribes asurface current of constant magnitude and rotating phase across thesurface of the antenna. Unfortunately, as metamaterial elements scatterwaves out of the antenna, the feed wave Hf decays and cannot exciteelements farther down the array as strongly as it can excite elements atthe beginning of the array. This effect depends entirely on how stronglythe elements couple to the feed wave and the decay in the feed waveincreases exponentially with an increase in coupling. Holographicmetamaterial antennas are therefore designed to operate away from theresonant frequency of the metamaterial elements in order to preventstrong coupling of the elements from decaying the feed wave too quickly.However, they also cannot operate far from the resonance due to the factthat if the elements couple too weakly, more energy is lost in thewaveguide and the overall efficiency will be reduced. Additionally, theindividual elements can only be tuned over a finite bandwidth, resultingin smaller variations in polarizability away from the resonance. As aresult, the operational bandwidth of a metasurface antenna is limiteddue to a minimum coupling level required to achieve a high efficiency.These two effects place a limitation on the usable bandwidth of theantenna.

To numerically investigate the broadband performance of the threediscussed modulation techniques, the far-field radiation patterns werecalculated across a large operational bandwidth. The tunablepolarizability for the idealized unit cells used to model the samplemetasurface antenna had a tuning range, and therefore operationalbandwidth, spanning 1 GHz (9.5-10.5 GHz). To compare the resultingfar-field beam patterns, the metrics of directivity and sidelobe levelwere extracted for each frequency and these results are plotted in FIG.6 and FIG. 7, respectively. From these plots, it is shown that while thebeam width is consistent for the three methods, the side-lobe levelsremain lower and more consistent in the case of Euclidean Modulation ascompared to the other methods.

Euclidean Modulation Processes

As discussed above, in one embodiment, Euclidean modulation uses the setof available polarizabilities to approximate the set of requiredpolarizabilities by minimizing the Euclidean distance between the two.An example of this is illustrated in FIG. 8, where each point on the setof required polarizabilities is mapped to the nearest point on theLorentzian curve. This mapping is given defined by the resonancefrequency ω_(0i) for the i^(th) metamaterial element located at positionr_(i) such that the distance

d=|α _(L)(ω_(0i))−α_(D)(r _(i))|  (14)

is a global minimum.

In one embodiment, minimizing the Euclidean distance between those twopoints comprises locally minimizing the error between the requiredpolarizability and the actual polarizability delivered by the dipole.This does not guarantee that the modulation pattern chosen by Euclideanmodulation is the globally best possible pattern, but it is an optimizedpattern in the sense that it is a local maximum of the directivityfunction.

FIG. 9A is a data flow diagram of one embodiment of a process forcontrolling an antenna. In one embodiment, the antenna comprises ametasurface antenna having a plurality of surface scattering antennaelements. Example embodiments of such an antenna are discussed in moredetail below. The proess is performed by processing logic that maycomprise hardware, software, firmware, or a combination of the three. Inone embodiment, the processing logic is part of a modem of the antenna.

Referring to FIG. 9A, the process begins with processing logicperforming a hologram calculation module (processing block 901). In oneembodiment, this calcuation is performed from equations 6a/b or 7a/b. Inone embodiment, the inputs for the hologram calculation include the wavepropagation in the feed, the location of the radiating cells, the beampointing direction and the polarization. The results of the hologramcalculation is the desired modulation.

Processing logic compares the desired modulation with the achievablemodulation states and uses them to perform a Euclidean modulation module(processing block 902). As discussed herein, the Euclidean modulationmodule maps the desired modulation to the available, or achievable,states. In one embodiment, the mapping of the desired modulation to theavailable states includes creating a resonator model, extracting compleximpedance values, and then mapping the complex desired modulation valuesto those impedance values based on Euclidean distance. In oneembodiment, the achievable states are calculated using an idealLorentzian approximation for the magnetic dipole. In an alternativeembodiment, a more realistic model of the resonator is generated.

The result of processing logic performing the Euclidean modulationmodule is the final modulation. Processing logic provides the finalmodulation to the antenna hardware to control the antenna to performbeamforming (processing block 903). In one embodiment, performing thebeamforming with the antenna hardware using the final modulationincludes mapping the final modulation values to control parameters(etc., control voltages to control, for example, the TFTs, diode currentif using varactor diodes to control element capacitance, etc.) andcontrolling RF radiating antenna element unit cells (e.g., surfacescattering antenna elements) to achieve the desired beamforming. In noeembodiment, a control board receives the modulation values as input andgenerates outputs that drive the row and column lines of the TFT array.

FIG. 9B is a flow diagram of one embodiment of a modulation process forcontrolling an antenna. The proess is performed by processing logic thatmay comprise hardware, software, firmware, or a combination of thethree.

Referring to FIG. 9B, the process begins by mapping a desired modulationto achievable modulation states (processing block 911). In oneembodiment, the desired modulation is obtained based on location of atleast a subset of the RF radiating antenna elements, beam pointingdirection and polarization, as well as the wave propagation in a feed ofthe antenna.

In one embodiment, the mapping a desired modulation to achievablemodulation states is based on Euclidian distance. In one embodiment, theRF radiating antenna elements comprise tunable elements in a metasurfaceand mapping the desired modulation to achievable modulation statescomprises approximating a set of required polarizabilities with a set ofthe tunable elements in the metasurface.

In one embodiment, mapping the desired modulation to achievablemodulation states comprises selecting points out of achievablepolarizabilities that approximate required polarizabilities of thedesired modulation. In one embodiment, selecting points out of a set ofachievable polarizabilities that approximate required polarizabilitiesof the desired modulation comprises minimizing distance between therequired and achievable polarizabilities. In one embodiment, minimizingdistance between the required and achievable polarizabilities comprisesminimizing a Euclidean norm between the required and achievablepolarizabilities.

After mapping a desired modulation to achievable modulation states,processing logic maps modulation values associated with the achievablemodulation states to one or more control parameters (processing block912). In one embodiment, the one or more control parameters comprise avoltage to be applied to each of the RF radiating antenna elements.

Using the control parameters, processing logic controls radio frequency(RF) radiating antenna elements of a metasurface (e.g., a metasurfaceantenna with surface scattering antenna elements such as described, forexample, in more detail below) to perform beam forming (processing block913).

In one embodiment, Euclidean modulation is implemented for ametamaterial antenna using the following algorithm:

-   -   1) Find the range of polarizabilities available for the        metamaterial elements, given the tuning range of the elements.        In one embodiment, these polarizabilities follow a Lorentzian of        the form of equation (8).    -   2) For each metamaterial element in the waveguide, compute the        ideal polarizability. In one embodiment, the ideal        polarizability is computed using equations (6a) and (6b).    -   3) For each element, find the point on the range of available        polarizabilities that is the shortest distance in the complex        plane from the ideal polarizability.    -   4) Tune each element so that it operates with the polarizability        that was found in step 3.

This process is shown in FIG. 9C. In one embodiment, the operations ofthe process are performed by a control board. In another embodiment, theoperations of the process are performed by processing logic and a modem.

To examine the far-field performance of Euclidean modulation versusbinary and greyshade modulation patterns, consider a waveguide that isaligned along the x-axis with 150 complementary metamaterial elementsthat are placed a distance Λ=213.6 apart from each other, where λ is thefree space wavelength of light. The incident field is also assumed to beunperturbed by the scattering of the elements in the guide, so that thisstudy only examines the inherent benefits of each of the modulationpatterns, without including the aperture taper. The resulting far-fieldpatterns of the three modulation patterns can be found using a standardarray factor calculation, and these are illustrated in FIGS. 10A, 10B,and 10C. The Euclidean modulation pattern removes most of the side-lobesthat are produced by the greyshade and binary patterns, and alsoexhibits a slightly higher maximum directivity of the main beam.

In one embodiment, the use of Euclidean modulation maintains an evenaperture illumination over a larger bandwidth than greyshade modulation,and therefore it offers superior broadband directivity. When a feed-wave(E_(μ), H_(μ)) travels through a leaky waveguide, the propagationconstant of the feed wave β is perturbed by the metamaterial elements onthe surface of the guide. If the metamaterial elements are spaceduniformly with spacing Λ down the guide, and all have the same magneticpolarizability α(ω), then the effect is that the feed-wave travels as ifit were through a homogenous medium with a new propagation constant

$\begin{matrix}{\beta_{eff} = {\beta + {\frac{\omega \; Z_{\mu}}{4\Lambda}{H_{\mu}^{2}\left( r_{i} \right)}{\alpha (\omega)}}}} & (11)\end{matrix}$

where H_(μ)(r_(i)) is the magnetic field of the mode at the location ofthe metamaterial elements, and Z_(μ) is a normalization constant definedby the Poynting vector of the waveguide mode

$\begin{matrix}{\frac{1}{Z_{\mu}} = {\int{\left( {E_{\mu} \times H_{\mu}} \right) \cdot {{da}.}}}} & (12)\end{matrix}$

If the metamaterial elements are not all the same, but vary by someperiodic modulation pattern then the average polarizability can be usedin equation (11). Based on equation (11), if the polarizability ismostly imaginary, then the perturbation of the propagation constant willmostly contribute to the absorption of the feed wave. If it is real andnegative, which is the case when the system operates above the resonancefrequency of the elements, then it will slow down the wave. If it isreal and positive, which is the case when the system operates below theresonance frequency of the elements, then it will speed up the wave.Slowing down or speeding up the feed wave can negatively impact theperformance of a holographic metamaterial antenna, since the modulationpattern is designed with a particular value for the propagation constantin mind. Comparing FIGS. 10A and 10B, the real part of the averagepolarizability for greyshade modulation will be negative if themodulation pattern is implemented below resonance, and this will slowdown the feed wave. However, since the real part of the averagepolarizability for Euclidean modulation is zero, Euclidean patterns willnot perturb the speed of the feed wave.

However, one effect on the propagation constant of the feed wave is thedecay due to the imaginary part of the polarizability. To examine thebandwidth of the system, consider an example system with 100metamaterial elements that is designed to operate at 18.7 GHz. Applyinga Euclidean modulation pattern defines a resonance frequency ω_(0i) foreach metamaterial element such that the polarizability of the i^(th)element, α_(i)=α(ω_(0i), ω), evaluated at the operating frequency ω=2πf,is as close as possible to the ideal polarizability. In FIG. 10D, thehistogram of the resonance frequencies of all the elements is plottedfor the Euclidean modulation pattern, which illustrates the density ofelements ρ(ω_(0i)) per unit frequency. Since the elements absorbsignificantly more when they operate on resonance than they do offresonance, the feed wave will decay much more rapidly when the antennaoperates at points in the frequency spectrum when the density ofelements with resonance frequency ω_(0i)≈ω than at a point in frequencywhere there are relatively few elements that are on resonance. There aretwo points of high density of elements, above and below the operatingfrequency. These points naturally arise in Euclidean modulation becausethe lower half plane in FIG. 8 mostly maps to elements that are turnedoff, which implies tuning them far away from the resonance frequency.

Using this density function, ρ(ω_(0i)), that defines the number ofmetamaterial elements with resonance frequency near some value ω_(0i),the average polarizability of the lattice of elements can be clearlydefined using the equation

$\begin{matrix}{< {\alpha (\omega)}>={\frac{1}{N}{\sum\limits_{i = 1}^{N}{{\rho \left( \omega_{0i} \right)}{\alpha \left( {\omega_{0i},\omega} \right)}}}}} & (17)\end{matrix}$

where α(ω_(0i), ω) is defined in equation (8). Using this expression forthe average polarizability in equation (11), the decay in the feed wavecan be effectively evaluated. Once the decay in the feed-wave is known,the directivity of the beam can be computed using a standard arrayfactor calculation, and the end-of-guide loss can be computed usingP_(L)=|S₂₁|², where S₂₁=e^(iβ) ^(eff) ^(Λ(N−1)). The directivity andend-of-guide loss are both shown in FIG. 11A, which shows that theantenna has a useable bandwidth of around 500 MHz. When the operatingfrequency deviates too far from the design frequency, the metamaterialstarts to absorb the feed wave more strongly, and hence the end of guideloss drops rapidly, as shown in FIG. 11A with return loss (red) andend-of-guide loss (blue). Unfortunately, even though the end-of-guideloss is decreasing, the directivity is also decreasing because theantenna can no longer sustain an even aperture. FIG. 11B illustratesmaximum directivity over frequency. The directivity of the antennaremains high for a large bandwidth centered around the operatingfrequency.

In comparison with prior art, a bandwidth of 500 MHz is significantlylarger than the bandwidth the system would exhibit under greyshademodulation. In FIG. 11C, the density of metamaterial element resonancefrequencies is plotted for this system using greyshade modulation. Inthis case, all of the resonance frequencies are clumped together in onesingle hump that is just below the operating frequency. This spot ofhigh density of elements means that the feed wave will decay veryrapidly in that region, and so the directivity will suffer, as shown inFIG. 11E. However, if the operating frequency is increased to higherfrequencies , then there are no more elements that resonate anywherenear the operating frequency, and the end of guide losses increase asshown in FIG. 11D. Because the resonance frequencies are clumpedtogether in a single localized area in greyshade modulation, thevariation in both the end-of-guide losses and the directivity overfrequency is much more severe than in Euclidean modulation, and thislimits the operating bandwidth of this system to be closer to 150 MHz,as opposed to 500 MHz in the case of Euclidean modulation.

Modulation Pattern Restriction

One of the common problems that plagues holographic metamaterialantennas is the decay of the feed wave. As the feed wave travels down ametamaterial waveguide, the feed wave is scattered by the metamaterialelements. If the elements are spaced a sub-wavelength distance apart,then the feed wave scatters in such a way that it obtains a new,effective propagation constant, which is given in equation (11).Unfortunately, if the metamaterial elements are radiative, then thiseffective propagation constant is complex, and so the primary effect isthat the feed wave decays exponentially, which creates an unevenaperture illumination that destroys the quality of the beam. However, inone embodiment, the polarizability of the metamaterial elements is tunedin such a way that the coupling of the elements increases down thelength of the guide, and the increase in coupling compensates for thedecay of the feed wave so that the magnitude of the dipole moment of allthe metamaterial elements is constant across the surface of the antenna.When the magnitude of the dipole moment of all the elements is the same,then all the elements radiate the same amount, and therefore theaperture is evenly illuminated and equation (2) will be satisfied,yielding a tight beam radiation pattern. Thus, modulation patternrestriction is a specific way to control the average coupling level ofthe metamaterial elements using the modulation pattern itself.

Required Average Polarizability for Even Aperture Illumination

Imagine the simple case where a rectangular waveguide with a width of band a height of a supports a single propagating mode that is propagatingalong the z-axis, and all the metamaterial elements have only oneorientation: in the {circumflex over (v)} direction. If this effectivepropagation constant is used to compute the required polarizabilitiesusing equations (7a) and (7b), then a transcendental equation for thepolarizability is obtained. The required polarization is the solution tothe equation,

$\begin{matrix}{\alpha_{i} = {{\frac{i\; \mu_{0}\Lambda^{2}Z_{ant}}{{kZ}_{0}}e^{{i{({k - \beta})}}y_{i}}e^{{iC} < {\alpha {(z)}} > {kz}_{i}}\mspace{14mu} {where}\mspace{14mu} C} = \left. {\frac{{cZ}_{\mu}}{4\Lambda}H_{\mu}^{2}} \right|_{z = h}}} & (13)\end{matrix}$

describes the coupling of the metamaterial elements to the waveguidemode, and the angle brackets denote local averaging. If the waveguide isrectangular with width b and height a, then C=√{square root over(1−(π/ka)²)}/(2abΛμ₀). The required average polarizability can becomputed from this equation by taking the local average of both sides ofthe equation. The local average may be defined using a slowly varyingenvelope approximation, such that α_(i)≡α(z_(i))≈<α(z)>e^(i(k−β)z) ^(i). This approximation is valid as long as the distance between theelements, Λ, is much less than the periodicity of the modulationpattern, which is defined by the factor e^(i(k−β)y) ^(i) . The period ofthe modulation pattern is L=2π/|k−β|, and so this approximation is validwhen Λ<<2π/|k−β|. Using this definition of the local average, theaverage polarizability is given by

(−ik/∇ ²μ₀)<α(z)>+(Z _(ant) /Z ₀)e ^(−iC<α(z)>kz)=0   (14)

which yields a transcendental equation that can be numerically solvedfor <α(z)>. Equation (14) gives a solution for how the averagepolarizability is to vary along the length of the waveguide in order tocompensate for the decay of the feed wave. Therefore, solving thistranscendental equation yields a completely self-consistent solution andnot a perturbative solution, in the sense that it takes into account thenew decay of the feed wave as the coupling level is modified. Thesolution of equation (14) is a purely imaginary average polarizability,which is guaranteed to be the case if the modulation pattern isEuclidean. The required imaginary part is plotted in FIG. 12 for arectangular waveguide with 100 metamaterial elements, and parametersa=13.5 mm, b=27 mm, w=2π10⁹ rad/s, Z_(ant)/Z₀=0.065, and Λ=λ/8. Thecoupling starts low at some level, and then increases with a veryspecific curve in order to guarantee an even aperture distribution.

Controlling the Average Coupling Level Using Restricted ModulationPatterns

In order to obtain an even aperture and good beam performance, theaverage coupling level of the metamaterial elements is controlled. Inone embodiment, the average coupling level is controlled by choosing anew map between the ideal polarizabilities and the range of possiblepolarizabilities that excludes any polarizability out of the range ofpossible polarizabilities that has an imaginary part that is greaterthan a certain level. Removing a certain portion of the range ofpossible polarizabilities on the Lorentzian curve is called restriction.Once the set of of possible polarizabilities is restricted to remove anypolarizabilities with an imaginary part greater than a certain level, aEuclidean modulation scheme may be applied to find the polarizabilityout of the remaining possible polarizabilities that is nearest in thecomplex plane to the required polarizability. Restricting the modulationpattern in this way introduces some additional error in the modulationpattern, since some phases that are achievable will be excluded.However, this phase error is usually very small in comparison with thebenefits gained by obtaining an even aperture distribution. FIGS. 13A-Dillustate altering Euclidean modulation by restriction. In oneembodiment, the pattern is restricted to only utilize region 2101 of theLorentzian curve to points where (a) α _(max)=α_(max), (b) α_(max)=(¾)α_(max), (c) α _(max)=(½)α_(max), (d) α _(max)=(¼)α_(max).

In one embodiment, in order to know what maximum imaginarypolarizability should be allowed by the modulation pattern in order toobtain a particular average polarizability, the average of the imaginarypart of the polarizability is computed for a Euclidean modulationpattern. For convenience, the following is defined

$\begin{matrix}{\alpha_{\max} = {\max\limits_{\omega_{0}}\left( {{Im}\left\{ \alpha_{L} \right\}} \right)}} & (15)\end{matrix}$

to be the maximum imaginary polarizability possible for some particularmetamaterial element, and ā_(max) to be the chosen maximum allowedimaginary polarizability for the modulation pattern to choose. If theEuclidean modulation pattern is restricted such that only points withthe imaginary part of the polarizability less than a certain value areallowed, then the average imaginary part of the polarizability of thepattern decreases. This effect is illustrated in FIGS. 13A-D, where, foreach point of desired polarizability, the nearest point on theLorentzian polarizability curve is found that has an imaginary part ofthe polarizability that is less than some value.

In FIGS. 13A-D, the effective impedance of the antenna for themodulation pattern is Z_(ant)=0.65Z₀, and the amplitude (or effectivesize) of the dipoles is set at A=0.2 mm³. Choosing these values for thesize of the dipoles and the impedance of the antenna fixes the maximumof the magnitude of the available polarizability equal to the magnitudeof the required polarizability, which minimizes the phase error and bestapproximates the curve of required polarizabilities with the curve ofavailable polarizabilities. Note that this value for Z_(ant) that isused to find the modulation pattern is different from the value ofZ_(ant) that is used to find the curve of needed average imaginarypolarizability to obtain an even aperture. This is ultimately becausethe average magnitude of the polarizability will be much smaller afterthe modulation pattern is applied than the ideal polarizability, becauseroughly half of the elements will end up with a polarizability that isnear zero. FIGS. 13A-D shows that reducing the maximum allowed imaginarypolarizability results in an average polarizability of <Im{α}>/μ₀=15.9in FIG. 13A, <Im{α}>/μ₀=14.6 mm³ in FIG. 13B, <Im{α}>/μ₀=11.8mm³ in FIG.13C, and <Im{α}>/μ₀=7.2 mm³ in FIG. 13D.

Based on this method of restriction, the average imaginary part of thepolarizability of the modulation pattern can be computed as a continuousfunction of the maximum allowed imaginary polarizability, which is shownin FIG. 14. Referring to FIG. 14, a calculation of the imaginary part ofthe average polarizability <Im{α}> is calculated as a function of themaximum allowed imaginary polarizability α _(max) for a restrictedEuclidean modulation pattern applied to metamaterial elements with A=0.2mm³. Once the function is known, it may be inverted to find α _(max) interms of (Im{α}), which may be used together with the results of solvingequation (14) to solve for the level of restriction of the modulationpattern, i.e., α _(max), that should be applied to each metamaterialelement down the length of the waveguide to obtain an even aperture.FIG. 15 illustrates a plot of the required restriction, α _(max), whichis required to obtain a particular average imaginary polarizability,(Im{α}) , that will result in an even aperture for the waveguidedescribed in FIG. 12.

At some point, as more and more of the feed wave decays and the couplinglevel increases, the coupling of the metamaterial elements reaches themaximum amount available (i.e., there is no restriction of the pattern)and so the available coupling is not able to keep up with the neededcoupling level to obtain an even aperture. At this point, we simplychoose to maintain the maximum coupling level all the way to the end ofthe guide.

Based on the above description, in one embodiment, the followingalgorithm implements restricted Euclidean modulation.

-   -   1) For each metamaterial element in the waveguide, compute the        ideal polarizability. In one embodiment, the ideal        polarizability is given by equations (6a) and (6b).    -   2) Find the required average polarizability distribution        required in order to obtain an even aperture for the        metamaterial elements. In one embodiment, this is performed by        numerically solving equation (14).    -   3) Find the range of polarizabilities available for the        metamaterial elements, given the tuning range of the elements.        In one embodiment, these polarizabilities follow a Lorentzian of        the form of equation (8).    -   4) Find the average imaginary polarizability as a function of        the maximum allowed imaginary polarizability. In one embodiment,        the average imaginary polarizability is set forth in equation        15, and is found using the range of available polarizabilities        found in step 3.    -   5) For each metamaterial element, find the maximum allowed        imaginary polarizability that is required in order to obtain an        even aperture. In one embodiment, this is found by inverting the        relationship found in step 4.    -   6) For each element, remove from its range of available        polarizabilities any points that have a larger imaginary        polarizability than the maximum allowed imaginary polarizability        that was perscribed by step 5. After those points are excluded,        this set will be defined as the new set of available        polarizabilities for the element.    -   7) For each element, find the point on the new range of        available polarizabilities, which was given by step 6, that is        the shortest distance in the complex plane from the ideal        polarizability found in step 1.    -   8) Tune each element so that it operates with the polarizability        that was found in step 7.

This above process is shown in FIG. 9D. In one embodiment, theoperations of the process are performed by a logic circuit such as aprocessor, microcontroller or an FPGA.

In one embodiment, the restricted Euclidean modulation creates an evenaperture excitation. As was mentioned above, restricting the pattern inthis manner does introduce some small phase error relative tounrestricted Euclidean modulation, but this phase error is generallyless costly than the benefits of obtaining an even aperturedistribution. These effects are shown in the far-field patterns. Inorder to accurately test the restriction method of obtaining an evenaperture, this antenna is modeled using the Discrete DipoleApproximation (DDA) that does not use effective medium averaging, butinstead takes into account the complete scattering of all the elementsinto the waveguide in a self-consistent manner. For more information onDDA, see M. Johnson, et al, “Discrete-dipole approximation model forcontrol and optimization of a holographic metamaterial antenna,” Appl.Opt., vol. 53, pp. 5791-5799, September 2014.

The final decay of the feed waves from modeling the antenna using theDDA under restricted and unrestricted Euclidean modulation are shown inFIG. 18A. The standard Euclidean modulation has the expected exponentialfeed wave decay, while restricted Euclidean has a decay profile thatlooks much more linear. The decay of the feed wave in the restrictedEuclidean pattern is in fact a multiple-exponential decay that beginswith a very slow decay rate, but increases its decay rate as it movesfarther down the guide.

In FIG. 18B, the polarizabilities of all the metamaterial elements downthe length of the waveguide are plotted for both restricted andunrestricted Euclidean modulation. The pattern oscillates rapidly, asexpected, for both patterns. However, under the restricted modulation,the average coupling level of the elements increases rapidly towards theend of the waveguide, while it stays constant for unrestricted Euclideanmodulation.

When the polarizability is multiplied by the feed wave amplitude, theresult is the magnitude of the dipole moments, or excitation level, ofeach of the metamaterial elements. This is plotted in FIG. 18A, which isthe product of FIG. 18B and FIG. 18C. Creating an aperture with cellswhere the average magnetic dipole moment is a constant is referred to asan even aperture, which is required by Schelkunoffs equivalenceprinciple in equation (3) to create an ideal beam pattern.

The far-field patterns of the restricted and unrestricted Euclideanmodulation schemes are compared in FIG. 16, using the results of the DDAanalysis. Restricted Euclidean modulation results in a higher andnarrower main beam with more directivity than unrestricted Euclideanmodulation, which is due to the evenness of the aperture.

In comparison, FIG. 17 shows the analysis applied to the same antenna,but with the effective index of the waveguide modified. In this case,several large side lobes appear in the restricted Euclidean pattern.

Note that the examples above are discussed in conjunction with linearwaveguide antennas. The techniques described above are applicable toother types of antennas (e.g., cylindrically fed antennas). One or moreexamples of such waveguides that may be used are disclosed below. Notethat for different antennas, the formulas above may need to be adjustedto use the propogation constant β of the particular waveguide and may bechanged to reflect the correct mode.

Bandwidth Adjustable Euclidean Modulation

In one embodiment, the Euclidean modulation that is used to select themodulation for use with the satellite antennas in a satellitecommunication system comprises bandwidth adjustable Euclideanmodulation. In one embodiment, the bandwidth adjustable Euclideanmodulation techniques described herein include performing a trade-offbetween gain and instantaneous bandwidth (IBW). That is, embodimentsdescribed herein include performing Euclidean modulation in a mannerthat makes a trade-off of gain for improved IBW. In one embodiment, theEuclidean modulation process makes adjustments to obtain improvements inIBW only in the software as opposed to relying on hardware modificationssuch as feed changes or scattering element changes. This helps enablethe techniques described herein to be useful in various applications,such as for example, but not limited to: (1) adjustable IBW, (2)interoperability between GEO and LEO, (3) front-end analog equalizer,(4) frequency filter to reject interference, and etc. These aredescribed in greater detail below.

As discussed above, IBW is defined herein as the frequency range thatthe antenna demonstrates consistent radiation characteristics withoutchanges in the modulation pattern. As the desired impedance for eachscattering antenna elements (RF radiating antenna elements) forming aholographic metasurface antenna is calculated using a holographicequation that relates the feed wave to a desired radiation pattern asdescribed above, and these impedance values are ideal and need to betranslated to an impedance that can be realized using the availablescattering antenna elements, Euclidean mapping is used to reduce, andpotentially minimize, the Euclidean distance (e.g., mapping error)between the desired and achievable impedances.

As discussed above, one way to do this calculation to reduce theEuclidean distance is at a single desired frequency, which can be, forexample, the center frequency, the modulation frequency, or the carrierfrequency. This means that the calculated modulation pattern isoptimized only for that frequency and not necessarily optimized at anyother frequency. Because of this, as the frequency moves further awayfrom that desired frequency, mapping errors can result in an offset inthe direction of the beam compared to the beam at the single desired orreference frequency, which is referred to as beam squinting.

In one embodiment, the Euclidean modulation applied is modified in thatit brings the frequency characteristics of the scattering antennaelements (e.g., the available impedances of the scattering elements)into calculation to reduce, and potentially minimize, the mapping errorwithin a bandwidth as opposed to a single frequency. In this way, thebeam squinting can be reduced to some extent and instantaneous bandwidthcan be improved. More specifically, both ideal and available impedancesare frequency dependent, so as the frequency changes from modulationfrequency, those values change. FIGS. 19 and 20 show an example of thisbehavior. In FIG. 19, the ‘o’ shows the ideal impedances. Observed thatas the frequency changes, the location of the ‘o’ also changes. Also inFIG. 19, the ‘+’ are the available impedances. As frequency changes, thelocation of each ‘+’ also moves. Therefore, know impedances vary versusfrequency is needed to make a decision on the mapping, which is doneusing the bandwidth adjustable Euclidean modulation disclosed herein.FIG. 20 shows an example and comparison with the conventional Euclideanmodulation.

The frequency bandwidth in which the characteristics of antenna elementsare taken into the calculation (referred to herein as a calculation orfrequency window) can be considered as a main trade-off parameter. Forexample, in one embodiment, the calculation window is centered around areference frequency, which may be the single reference modulationfrequency. Alternatively, the calculation window is centered around adifferent reference frequency. In yet another alternative, the referencefrequency represents an endpoint such that the calculation windowexpands in a single frequency (increasing or decreasing). That is, thecalculation window can be centered around the modulation frequency orshifted. There are cases where the gain curve versus frequency (e.g.,the solid line in FIG. 21) falls faster at one side compared to theother side. The shifting of calculation window acts as an equalizingmechanism to improve, and potentially optimize, the modulation for oneside more than the other side. In this way, a flatter gain curve can beachieved.

As the size of the calculation window increases, the instantaneousbandwidth is expected to improve at the expense of gain reduction at themodulation frequency. The reason for this reduction is that themodulation will no longer be optimized for that frequency.

FIG. 19 shows the example where the Euclidean mapping calculation isdone only at the modulation frequency, which in the example is 10.4 GHz.While the mapping is optimized at that frequency, it is not optimized(i.e., mapping error is not minimum) at other frequencies, such as, forexample, at 10.7 GHz.

FIG. 20 shows an example of when a calculation window is applied toaccount for a range of frequency characteristics of the elements.Referring to FIG. 20, although the mapping is slightly adjusted at 10.4GHz, the mapping error is significantly reduced at 10.7 GHz.

The operation of bandwidth adjustable Euclidean modulation was tested ona radial metasurface antenna with tunable elements in the far-fieldchamber. FIG. 21 shows a comparison of gain achieved for traditionalEuclidean modulation at the reference frequency (10.4 GHz) and gainachieved for bandwidth adjustable Euclidean modulation with an examplecalculation window. Referring to FIG. 21, an example of the trade-offbetween gain and instantaneous bandwidth (IBW) is shown. IBW here isdefined as the gain flatness (peak-to-peak variation) within 230.4 MHzaround the modulation frequency (±115.2 MHz). The solid curve shows thegain vs. frequency curve at 60 deg scan where the modulation patternbased on traditional Euclidean modulation is applied at 10.4 GHz. Thedashed curve shows the gain curve where the modulation pattern based onbandwidth adjustable Euclidean modulation is applied. In this case, theacceptable gain drop was set to be less than 0.5 dB. Note that thepeak-to-peak variation can be significantly improved at the expense ofslight gain reduction.

FIG. 22A is a flow diagram of one embodiment of a process for performingbandwidth adjustable Euclidean modulation processing and optimization.The process is performed by processing logic that may comprise hardware(circuitry, dedicated logic, etc.), software (such as is run on ageneral-purpose computer system, a server, or a dedicated machine),firmware, or a combination of the three. In one embodiment, theprocessing logic is part of modulation logic in a modulator and/ordemodulator of an antenna is a satellite terminal.

Referring to FIG. 22A, the process begins by processing logic determinesthe radiation characteristics of a flat-panel metasurface antenna(processing block 2201). In one embodiment, the radiationcharacteristics comprise frequency and beam characteristics indicatingwhere the beam of the antenna is pointing. In one embodiment, theradiation characteristics comprise the scan angle and the linearpolarization angle. In one embodiment, the beam characteristics includethe propagation direction (e.g., specified by Theta/Phi, orAzimuth/Elevation in spherical coordinate system, etc.) and the linearpolarization angle, where the linear polarization angle describes theangle difference between the co-polarized field component and thevertical field component (E_theta).

In one embodiment, the process considers the radiation requirements forlow-earth orbit (LEO)-based transmitting devices, mid-earth orbit(MEO)-based transmitting devices, or geostationary transmitting devices.These requirements can be dynamic in nature in terms of how data istransmitted and there are many differences in the requirements, mainlyreferring to gain and IBW in this case. For instance, LEO devices mayrequire lower gain, but larger bandwidth compared to GEO devices.

Based on the characteristics, processing logic calculates or obtainsideal impedance values for all the antenna elements (e.g., RF radiatingelements of a metasurface, etc.) using the holographic equations (2202)as inputs for determining the ideal impedances (processing block 2203).In one embodiment, this is performed in a manner well-known in the art.

Processing logic then determines the available impedance values of theelements (processing block 2204). In one embodiment, the values arepredetermined and stored in memory and accessed by the modulationprocess.

Initially, processing logic translates the ideal impedances into actualimpedances using the traditional Euclidean modulation (described above)and tests whether the comparison indicates whether the gain issufficient to allow for adjustments (e.g., is the gain associated withthe modulation pattern above a minimum threshold) (processing block2205). In one embodiment, the gain requirement is specified is set forthe entire RF chain, and the RF chain budget determines the requirementon the antenna gain..

If there is insufficient gain to meet requirements, the processterminates as a further adjustment based on instantaneous bandwidthwould result in insufficient gain. However, if there is sufficient gain,the process transitions to processing block 2206 where processing logicdetermines whether the IBW requirements have been met previously. In oneembodiment, the IBW requirements are set for the entire RF chain, andthe RF chain budget determines the requirement on the antenna IBW. Ifthe IBW requirements have been met, then the process ends. If the IBWrequirements have not been met, then the process transitions toprocessing block 2207 where the calculation window width is increasedand the bandwidth adjustment is made in the Euclidean modulationprocess. In the process flow example, a default initial calculationwindow is set at 0 MHz. In one embodiment, the amount at each iterationcan be fixed and set by a user. In another embodiment, this can be doneusing a search algorithm where the amount can vary at each iteration.For example, if targeting +/−125 MHz IBW, the calculation windows of 0to +/−240 Mhz may be examined in the steps of 20 MHz.

In one embodiment, the determination requirements for the calculationwindow are based on the radiation requirement of the metasurfaceantenna. In the example in FIG. 22A, these requirements are antenna gainand instantaneous bandwidth. These requirements can change to meetvarious anticipated scenarios of the application of the metasurfaceantenna. For example, the radiation characteristics for LEO-basedMEO-based or geostationary transmitting devices can determine thedesired or optimal range of the calculation window. In other examples,the requirements and subsequently, the calculation window can be basedon other factors such as, for example, but not limited to, aging,temperature variations, atmospheric conditions, etc.

As discussed above, thus, if the instantaneous bandwidth requirement ismet by the existing calculation window, the process can end. If not, theprocess will proceed to adjust the calculation window and repeat thecomparison of the radiation characteristics with the requirements. Thiscontinues until the further adjustments in gain are available and theIBW requirements are met. More specifically, as Gain is defined at asingle frequency (i.e., modulation frequency) and IBW is defined as gainvariation vs frequency, the gain data at frequencies other than themodulation frequency is needed. An example of requirement ispeak-to-peak gain variation less than 1 dB in 250 MHz. In this case, thegain versus frequency within +/−125 MHz of the modulation frequency isexamined, and if the gain variation is less than 1 dB, then therequirement is met.

FIG. 22B is a flow diagram of another embodiment of a process forperforming bandwidth adjustable Euclidean modulation. The process isperformed by processing logic that may comprise hardware (circuitry,dedicated logic, etc.), software (such as is run on a general-purposecomputer system, a server, or a dedicated machine), firmware, or acombination of the three. In one embodiment, the processing logic ispart of modulation logic in a modulator and/or demodulator of an antennais a satellite terminal.

Referring to FIG. 22B, the process begins by processing logic selectinga modulation for a frequency range to be applied to antenna elements(e.g., surface scattering antenna elements) of an aperture of an antenna(e.g., satellite antenna) (processing block 2211). In one embodiment,the antenna elements comprises RF radiating antenna elements. In oneembodiment, the RF radiating antenna elements comprise tunable elementsin a metasurface.

In one embodiment, performing modulation for a frequency range occurs inresponse to determining that instantaneous bandwidth (IBW) requirementsare not met by a beam formed by the antenna with another set ofmodulation values associated with another, previously selected frequencyrange. The previously selected frequency range may consist of a singlefrequency or a frequency band that includes multiple frequency values.For example, in one embodiment, on a first pass through the process, thefrequency range is a single frequency. In the second or any later passthrough the process, the frequency range includes bandwidth.

In one embodiment, determining instantaneous bandwidth (IBW)requirements are not met by a beam formed by the antenna with anotherset of modulation values associated with a previously selected frequencyrange is performed after determining gain is sufficient to allow foradjustments to the frequency range associated with that previously setof modulation values in order to satisfy the IBW requirements.

In one embodiment, selecting a modulation comprises obtaining thedesired modulation, mapping the desired modulation to the achievablemodulation states, and determining a first set of modulation values forachievable modulation states associated with the frequency range. In oneembodiment, the desired modulation is based on one or more of: locationof at least a subset of the RF radiating antenna elements, beam pointingdirection and polarization, and wave propagation in a feed of theantenna. In one embodiment, mapping the desired modulation to theachievable modulation states is based on Euclidian distance betweenrequired and achievable polarizabilities and each of the achievablemodulation states corresponds to voltages applied to the antennaelements of the antenna to induce magnetic dipole moments withachievable polarizabilities. In one embodiment, where the antennaelements comprises the RF radiating antenna elements comprise tunableelements in a metasurface and mapping the desired modulation to theachievable modulation states comprises approximating a set of requiredpolarizabilities with a set of the tunable elements in the metasurface.

After selecting the modulation for the frequency range, processing logicmaps modulation values associated with the achievable modulation statesto one or more control parameters (processing block 2212) and controlsthe radiating antenna elements using the one or more control parametersto perform beam forming (processing block 2213). In one embodiment,these control parameters comprise a voltage to be applied to each of theRF radiating antenna elements.

FIG. 22C is a flow diagram of yet another embodiment of a process forperforming bandwidth adjustable Euclidean modulation. The process isperformed by processing logic that may comprise hardware (circuitry,dedicated logic, etc.), software (such as is run on a general-purposecomputer system, a server, or a dedicated machine), firmware, or acombination of the three. In one embodiment, the processing logic ispart of modulation logic in a modulator and/or demodulator of an antennais a satellite terminal.

Referring to FIG. 22C, the process begins by processing logicdetermining, for a first frequency window (e.g., a single frequency, afrequency window/range), a desired modulation associated with a desiredimpedance for each radio frequency (RF) radiating antenna element of aplurality of RF radiating antenna elements of an antenna, using aholographic equation that relates a feed wave to a desired radiationpattern (processing block 2221).

Using this information, processing logic maps the desired modulation toachievable modulation states using Euclidean mapping (processing block2222) and determines the gain associated with a modulation pattern forthe frequency window (processing block 2223). In one embodiment, themapping of a desired modulation to achievable modulation states is basedon Euclidian distance. In one embodiment, the RF radiating antennaelements comprise tunable elements in a metasurface and mapping thedesired modulation to achievable modulation states comprisesapproximating a set of required polarizabilities with a set of thetunable elements in the metasurface.

Processing logic determines whether the gain associated with themodulation pattern is above a threshold (processing block 2224). In oneembodiment, the threshold is the gain requirement set by a satelliteservices provider. If the gain is not above the threshold, then theprocess ends (processing block 2225).

If the gain is above the threshold, then the process transitions toprocessing block 2226 where processing logic determines whetherrequirements for instantaneous bandwidth (IBW) have been met previouslyby the modulation pattern associated with the frequency window. If theIBW requirements are not met by the modulation pattern associatedfrequency window, then processing logic adjusts the first frequencywindow by increasing size of the frequency window to create a newfrequency window and repeats the process using the new frequency window(processing block 2227).

If the IBW requirements are met by the modulation pattern associatedfrequency window, then processing logic maps modulation valuesassociated with the achievable modulation states for the modulationpattern to one or more control parameters and controls radio frequency(RF) radiating antenna elements using the one or more control parametersto perform beam forming (processing block 2228). In one embodiment,these control parameters comprise a voltage to be applied to each of theRF radiating antenna elements.

Embodiments described herein include a number of additional innovationsand improvements besides those mentioned above. For example, in oneembodiment, the antenna dynamically adjusts the modulation pattern toaccount for various scenarios (e.g., rain fade). In one embodiment, asthe rain fade causes degradation in the signal, such that a higherantenna gain is needed to compensate, a reverse case could be used wherethe calculation window is already applied but the gain drops below athreshold due to the rain fade. In this scenario, in one embodiment, IBWis traded to increase the gain.

As another example, in one embodiment, the antenna dynamically adjuststhe IBW based on the data payload. For instance, the requirements mightbe different for LEO and GEO applications, so the antenna candynamically adjust and trade gain or G/T vs IBW. In such a case, ahigher G/T could be traded in favor of IBW if the IBW is not wideenough.

In one embodiment, Euclidean modulation is also performed in a mannerthat makes a trade-off for cost: Low Earth orbiting satellite (LEO)versus geostationary Earth orbit (GEO) (e.g., smart hub). Morespecifically, the ability to use bandwidth adjustable Euclideanmodulation allows an antenna to be configured to provide support forboth GEO and LEO and enable it for interoperability between LEO and GEO.In one embodiment, this is accomplished by increasing the diameter ofthe antenna to meet the GEO G/T requirements. However, to meet LEOinstantaneous bandwidth requirements with a larger aperture size, themodulation techniques disclosed herein in accordance with one embodimenttrade off gain and G/T for instantaneous bandwidth if a GEO terminal isupdated to turn it into a LEO terminal. Thus, in one embodiment, theinteroperability of the terminal may be dynamically adjusted to operatebetween GEO and LEO terminals. After such an adjustment, the modulationpattern is adjusted to communicate with either type of satellites.

In one embodiment, the techniques disclosed herein enable the antenna toact as a front-end analog equalizer, thereby potentially eliminating theneed for equalizers in the RF chain. Currently, equalizers are used toadjust the unequal gain within the bandwidth. These variations can bedue to the antenna bandwidth as well as the channel impairment thatwould produce unequal gain. In one embodiment, the adjustable bandwidthEuclidean modulation techniques disclosed herein assist the equalizer incompensation or replace it when its settings are adjusted using feedbackfrom the demodulator. In a broader point of view, this method can beused to automatically compensate for other factors such, for example,but not limited to aging, temperature variations, etc.

One way to implement enabling the antenna to act as a front-end analogequalizer is by training an adaptive equalizer by using either blindequalization or using a known signal such as the physical layer Start ofFrame of a waveform (e.g., DVB-S2X). Once the adaptive equalizer hasbeen trained, equalization is applied to specific antenna elements. Forexample, in one embodiment, the adaptive equalizer taps are interpolatedand then a similar pattern is applied to the antenna for that specificorientation. The antenna orientation is factored in since the adaptiveequalizer can be basically thought of as finding a set of delays thatreduce inter-symbol interference as well as other artifacts. In oneembodiment, an equalizer is still be used to estimate the delaycorrections required for a specific orientation.

In another embodiment, the techniques described herein enable theantenna to act as a frequency filter with adjustable bandwidth topotentially exclude and/or eliminate the interference when getting intothe passband. Referring to FIG. 21, for example, the gain curve vsfrequency can be interpreted as a filtering response where the antennabeam will be only directive within the vicinity of the modulationfrequency. In this perspective, the filtering response can become wideror narrower using the bandwidth adjustable Euclidean modulationtechniques disclosed herein. If the filtering response is wide and aninterference occurs within the band, the parameters (e.g., calculationwindow) in the bandwidth adjustable Euclidean modulation techniques canbe adjusted to narrow the filtering response in order toexclude/eliminate the interference.

Examples of Antenna Embodiments

The techniques described above may be used with flat panel antennas.Embodiments of such flat panel antennas are disclosed. The flat panelantennas include one or more arrays of antenna elements on an antennaaperture. In one embodiment, the antenna elements comprise liquidcrystal cells. In one embodiment, the flat panel antenna is acylindrically fed antenna that includes matrix drive circuitry touniquely address and drive each of the antenna elements that are notplaced in rows and columns. In one embodiment, the elements are placedin rings.

In one embodiment, the antenna aperture having the one or more arrays ofantenna elements is comprised of multiple segments coupled together.When coupled together, the combination of the segments form closedconcentric rings of antenna elements. In one embodiment, the concentricrings are concentric with respect to the antenna feed.

Overview of an Example of Antenna Systems

In one embodiment, the flat panel antenna is part of a metamaterialantenna system. Embodiments of a metamaterial antenna system forcommunications satellite earth stations are described. In oneembodiment, the antenna system is a component or subsystem of asatellite earth station (ES) operating on a mobile platform (e.g.,aeronautical, maritime, land, etc.) that operates using either Ka-bandfrequencies or Ku-band frequencies for civil commercial satellitecommunications. Note that embodiments of the antenna system also can beused in earth stations that are not on mobile platforms (e.g., fixed ortransportable earth stations).

In one embodiment, the antenna system uses surface scatteringmetamaterial technology to form and steer transmit and receive beamsthrough separate antennas. In one embodiment, the antenna systems areanalog systems, in contrast to antenna systems that employ digitalsignal processing to electrically form and steer beams (such as phasedarray antennas).

In one embodiment, the antenna system is comprised of three functionalsubsystems: (1) a wave guiding structure consisting of a cylindricalwave feed architecture; (2) an array of wave scattering metamaterialunit cells that are part of antenna elements; and (3) a controlstructure to command formation of an adjustable radiation field (beam)from the metamaterial scattering elements using holographic principles.

Examples of Wave Guiding Structures

FIG. 23 illustrates an aperture having one or more arrays of antennaelements placed in concentric rings around an input feed of thecylindrically fed antenna. In one embodiment, the cylindrically fedantenna includes a coaxial feed that is used to provide a cylindricalwave feed. In one embodiment, the cylindrical wave feed architecturefeeds the antenna from a central point with an excitation that spreadsoutward in a cylindrical manner from the feed point. That is, acylindrically fed antenna creates an outward travelling concentric feedwave. Even so, the shape of the cylindrical feed antenna around thecylindrical feed can be circular, square or any shape. In anotherembodiment, a cylindrically fed antenna creates an inward travellingfeed wave. In such a case, the feed wave most naturally comes from acircular structure.

Antenna Elements

In one embodiment, the antenna elements comprise a group of patchantennas. This group of patch antennas comprises an array of scatteringmetamaterial elements. In one embodiment, each scattering element in theantenna system is part of a unit cell that consists of a lowerconductor, a dielectric substrate and an upper conductor that embeds acomplementary electric inductive-capacitive resonator (“complementaryelectric LC” or “CELC”) that is etched in or deposited onto the upperconductor. As would be understood by those skilled in the art, LC in thecontext of CELC refers to inductance-capacitance, as opposed to liquidcrystal.

In one embodiment, a liquid crystal (LC) is disposed in the gap aroundthe scattering element. This LC is driven by the direct driveembodiments described above. In one embodiment, liquid crystal isencapsulated in each unit cell and separates the lower conductorassociated with a slot from an upper conductor associated with itspatch. Liquid crystal has a permittivity that is a function of theorientation of the molecules comprising the liquid crystal, and theorientation of the molecules (and thus the permittivity) can becontrolled by adjusting the bias voltage across the liquid crystal.Using this property, in one embodiment, the liquid crystal integrates anon/off switch for the transmission of energy from the guided wave to theCELC. When switched on, the CELC emits an electromagnetic wave like anelectrically small dipole antenna. Note that the teachings herein arenot limited to having a liquid crystal that operates in a binary fashionwith respect to energy transmission.

In one embodiment, the feed geometry of this antenna system allows theantenna elements to be positioned at forty-five-degree (45°) angles tothe vector of the wave in the wave feed. Note that other positions maybe used (e.g., at 40° angles). This position of the elements enablescontrol of the free space wave received by or transmitted/radiated fromthe elements. In one embodiment, the antenna elements are arranged withan inter-element spacing that is less than a free-space wavelength ofthe operating frequency of the antenna. For example, if there are fourscattering elements per wavelength, the elements in the 30 GHz transmitantenna will be approximately 2.5 mm (i.e., ¼th the 10 mm free-spacewavelength of 30 GHz). In one embodiment, the two sets of elements areperpendicular to each other and simultaneously have equal amplitudeexcitation if controlled to the same tuning state. Rotating them +/−45degrees relative to the feed wave excitation achieves both desiredfeatures at once. Rotating one set 0 degrees and the other 90 degreeswould achieve the perpendicular goal, but not the equal amplitudeexcitation goal. Note that 0 and 90 degrees may be used to achieveisolation when feeding the array of antenna elements in a singlestructure from two sides.

The amount of radiated power from each unit cell is controlled byapplying a voltage to the patch (potential across the LC channel) usinga controller. Traces to each patch are used to provide the voltage tothe patch antenna. The voltage is used to tune or detune the capacitanceand thus the resonance frequency of individual elements to effectuatebeam forming. The voltage required is dependent on the liquid crystalmixture being used. The voltage tuning characteristic of liquid crystalmixtures is mainly described by a threshold voltage at which the liquidcrystal starts to be affected by the voltage and the saturation voltage,above which an increase of the voltage does not cause major tuning inliquid crystal. These two characteristic parameters can change fordifferent liquid crystal mixtures.

In one embodiment, as discussed above, a matrix drive is used to applyvoltage to the patches in order to drive each cell separately from allthe other cells without having a separate connection for each cell(direct drive). Because of the high density of elements, the matrixdrive is an efficient way to address each cell individually.

In one embodiment, the control structure for the antenna system has 2main components: the antenna array controller, which includes driveelectronics, for the antenna system, is below the wave scatteringstructure, while the matrix drive switching array is interspersedthroughout the radiating RF array in such a way as to not interfere withthe radiation. In one embodiment, the drive electronics for the antennasystem comprise commercial off-the shelf LCD controls used in commercialtelevision appliances that adjust the bias voltage for each scatteringelement by adjusting the amplitude or duty cycle of an AC bias signal tothat element.

In one embodiment, the antenna array controller also contains amicroprocessor executing the software. The control structure may alsoincorporate sensors (e.g., a GPS receiver, a three-axis compass, a3-axis accelerometer, 3-axis gyro, 3-axis magnetometer, etc.) to providelocation and orientation information to the processor. The location andorientation information may be provided to the processor by othersystems in the earth station and/or may not be part of the antennasystem.

More specifically, the antenna array controller controls which elementsare turned off and those elements turned on and at which phase andamplitude level at the frequency of operation. The elements areselectively detuned for frequency operation by voltage application.

For transmission, a controller supplies an array of voltage signals tothe RF patches to create a modulation, or control pattern. The controlpattern causes the elements to be turned to different states. In oneembodiment, multistate control is used in which various elements areturned on and off to varying levels, further approximating a sinusoidalcontrol pattern, as opposed to a square wave (i.e., a sinusoid grayshade modulation pattern). In one embodiment, some elements radiate morestrongly than others, rather than some elements radiate and some do not.Variable radiation is achieved by applying specific voltage levels,which adjusts the liquid crystal permittivity to varying amounts,thereby detuning elements variably and causing some elements to radiatemore than others.

The generation of a focused beam by the metamaterial array of elementscan be explained by the phenomenon of constructive and destructiveinterference. Individual electromagnetic waves sum up (constructiveinterference) if they have the same phase when they meet in free spaceand waves cancel each other (destructive interference) if they are inopposite phase when they meet in free space. If the slots in a slottedantenna are positioned so that each successive slot is positioned at adifferent distance from the excitation point of the guided wave, thescattered wave from that element will have a different phase than thescattered wave of the previous slot. If the slots are spaced one quarterof a guided wavelength apart, each slot will scatter a wave with a onefourth phase delay from the previous slot.

Using the array, the number of patterns of constructive and destructiveinterference that can be produced can be increased so that beams can bepointed theoretically in any direction plus or minus ninety degrees(90°) from the bore sight of the antenna array, using the principles ofholography. Thus, by controlling which metamaterial unit cells areturned on or off (i.e., by changing the pattern of which cells areturned on and which cells are turned off), a different pattern ofconstructive and destructive interference can be produced, and theantenna can change the direction of the main beam. The time required toturn the unit cells on and off dictates the speed at which the beam canbe switched from one location to another location.

In one embodiment, the antenna system produces one steerable beam forthe uplink antenna and one steerable beam for the downlink antenna. Inone embodiment, the antenna system uses metamaterial technology toreceive beams and to decode signals from the satellite and to formtransmit beams that are directed toward the satellite. In oneembodiment, the antenna systems are analog systems, in contrast toantenna systems that employ digital signal processing to electricallyform and steer beams (such as phased array antennas). In one embodiment,the antenna system is considered a “surface” antenna that is planar andrelatively low profile, especially when compared to conventionalsatellite dish receivers.

FIG. 24 illustrates a perspective view of one row of antenna elementsthat includes a ground plane and a reconfigurable resonator layer.Reconfigurable resonator layer 1230 includes an array of tunable slots1210. The array of tunable slots 1210 can be configured to point theantenna in a desired direction. Each of the tunable slots can betuned/adjusted by varying a voltage across the liquid crystal.

Control module 1280 is coupled to reconfigurable resonator layer 1230 tomodulate the array of tunable slots 1210 by varying the voltage acrossthe liquid crystal in FIG. 25A. Control module 1280 may include a FieldProgrammable Gate Array (“FPGA”), a microprocessor, a controller,System-on-a-Chip (SoC), or other processing logic. In one embodiment,control module 1280 includes logic circuitry (e.g., multiplexer) todrive the array of tunable slots 1210. In one embodiment, control module1280 receives data that includes specifications for a holographicdiffraction pattern to be driven onto the array of tunable slots 1210.The holographic diffraction patterns may be generated in response to aspatial relationship between the antenna and a satellite so that theholographic diffraction pattern steers the downlink beams (and uplinkbeam if the antenna system performs transmit) in the appropriatedirection for communication. Although not drawn in each figure, acontrol module similar to control module 1280 may drive each array oftunable slots described in the figures of the disclosure.

Radio Frequency (“RF”) holography is also possible using analogoustechniques where a desired RF beam can be generated when an RF referencebeam encounters an RF holographic diffraction pattern. In the case ofsatellite communications, the reference beam is in the form of a feedwave, such as feed wave 1205 (approximately 20 GHz in some embodiments).To transform a feed wave into a radiated beam (either for transmittingor receiving purposes), an interference pattern is calculated betweenthe desired RF beam (the object beam) and the feed wave (the referencebeam). The interference pattern is driven onto the array of tunableslots 1210 as a diffraction pattern so that the feed wave is “steered”into the desired RF beam (having the desired shape and direction). Inother words, the feed wave encountering the holographic diffractionpattern “reconstructs” the object beam, which is formed according todesign requirements of the communication system. The holographicdiffraction pattern contains the excitation of each element and iscalculated by W_(hologram)=w_(in)*w_(out), with w_(in) as the waveequation in the waveguide and w_(out) the wave equation on the outgoingwave.

FIG. 25A illustrates one embodiment of a tunable resonator/slot 1210.Tunable slot 1210 includes an iris/slot 1212, a radiating patch 1211,and liquid crystal 1213 disposed between iris 1212 and patch 1211. Inone embodiment, radiating patch 1211 is co-located with iris 1212.

FIG. 25B illustrates a cross section view of one embodiment of aphysical antenna aperture. The antenna aperture includes ground plane1245, and a metal layer 1236 within iris layer 1233, which is includedin reconfigurable resonator layer 1230. In one embodiment, the antennaaperture of FIG. 25B includes a plurality of tunable resonator/slots1210 of FIG. 25A. Iris/slot 1212 is defined by openings in metal layer1236. A feed wave, such as feed wave 1205 of FIG. 24, may have amicrowave frequency compatible with satellite communication channels.The feed wave propagates between ground plane 1245 and resonator layer1230.

Reconfigurable resonator layer 1230 also includes gasket layer 1232 andpatch layer 1231. Gasket layer 1232 is disposed between patch layer 1231and iris layer 1233. Note that in one embodiment, a spacer could replacegasket layer 1232. In one embodiment, iris layer 1233 is a printedcircuit board (“PCB”) that includes a copper layer as metal layer 1236.In one embodiment, iris layer 1233 is glass. Iris layer 1233 may beother types of substrates.

Openings may be etched in the copper layer to form slots 1212. In oneembodiment, iris layer 1233 is conductively coupled by a conductivebonding layer to another structure (e.g., a waveguide) in FIG. 25B. Notethat in an embodiment the iris layer is not conductively coupled by aconductive bonding layer and is instead interfaced with a non-conductingbonding layer.

Patch layer 1231 may also be a PCB that includes metal as radiatingpatches 1211. In one embodiment, gasket layer 1232 includes spacers 1239that provide a mechanical standoff to define the dimension between metallayer 1236 and patch 1211. In one embodiment, the spacers are 75microns, but other sizes may be used (e.g., 3-200 mm). As mentionedabove, in one embodiment, the antenna aperture of FIG. 25B includesmultiple tunable resonator/slots, such as tunable resonator/slot 1210includes patch 1211, liquid crystal 1213, and iris 1212 of FIG. 25B. Thechamber for liquid crystal 1213 is defined by spacers 1239, iris layer1233 and metal layer 1236. When the chamber is filled with liquidcrystal, patch layer 1231 can be laminated onto spacers 1239 to sealliquid crystal within resonator layer 1230.

A voltage between patch layer 1231 and iris layer 1233 can be modulatedto tune the liquid crystal in the gap between the patch and the slots(e.g., tunable resonator/slot 1210). Adjusting the voltage across liquidcrystal 1213 varies the capacitance of a slot (e.g., tunableresonator/slot 1210). Accordingly, the reactance of a slot (e.g.,tunable resonator/slot 1210) can be varied by changing the capacitance.Resonant frequency of slot 1210 also changes according to the equation

$f = \frac{1}{2\pi \sqrt{LC}}$

where f is the resonant frequency of slot 1210 and L and C are theinductance and capacitance of slot 1210, respectively. The resonantfrequency of slot 1210 affects the energy radiated from feed wave 1205propagating through the waveguide. As an example, if fed wave 1205 is 20GHz, the resonant frequency of a slot 1210 may be adjusted (by varyingthe capacitance) to 17 GHz so that the slot 1210 couples substantiallyno energy from feed wave 1205. Or, the resonant frequency of a slot 1210may be adjusted to 20 GHz so that the slot 1210 couples energy from feedwave 1205 and radiates that energy into free space. Although theexamples given are binary (fully radiating or not radiating at all),full gray scale control of the reactance, and therefore the resonantfrequency of slot 1210 is possible with voltage variance over amulti-valued range. Hence, the energy radiated from each slot 1210 canbe finely controlled so that detailed holographic diffraction patternscan be formed by the array of tunable slots.

In one embodiment, tunable slots in a row are spaced from each other byλ/5. Other spacings may be used. In one embodiment, each tunable slot ina row is spaced from the closest tunable slot in an adjacent row by λ/2,and, thus, commonly oriented tunable slots in different rows are spacedby λ/4, though other spacings are possible (e.g., λ/5, λ/6.3). Inanother embodiment, each tunable slot in a row is spaced from theclosest tunable slot in an adjacent row by λ/3.

Embodiments use reconfigurable metamaterial technology, such asdescribed in U.S. patent application Ser. No. 14/550,178, entitled“Dynamic Polarization and Coupling Control from a SteerableCylindrically Fed Holographic Antenna”, filed Nov. 21, 2014 and U.S.patent application Ser. No. 14/610,502, entitled “Ridged Waveguide FeedStructures for Reconfigurable Antenna”, filed Jan. 30, 2015.

FIGS. 26A-D illustrate one embodiment of the different layers forcreating the slotted array. The antenna array includes antenna elementsthat are positioned in rings, such as the example rings shown in FIG.23. Note that in this example the antenna array has two different typesof antenna elements that are used for two different types of frequencybands.

FIG. 26A illustrates a portion of the first iris board layer withlocations corresponding to the slots. Referring to FIG. 26A, the circlesare open areas/slots in the metallization in the bottom side of the irissubstrate, and are for controlling the coupling of elements to the feed(the feed wave). Note that this layer is an optional layer and is notused in all designs. FIG. 26B illustrates a portion of the second irisboard layer containing slots. FIG. 26C illustrates patches over aportion of the second iris board layer. FIG. 26D illustrates a top viewof a portion of the slotted array.

FIG. 27 illustrates a side view of one embodiment of a cylindrically fedantenna structure. The antenna produces an inwardly travelling waveusing a double layer feed structure (i.e., two layers of a feedstructure). In one embodiment, the antenna includes a circular outershape, though this is not required. That is, non-circular inwardtravelling structures can be used. In one embodiment, the antennastructure in FIG. 27 includes the coaxial feed of FIG. 23.

Referring to FIG. 27, a coaxial pin 1601 is used to excite the field onthe lower level of the antenna. In one embodiment, coaxial pin 1601 is a50 Ω coax pin that is readily available. Coaxial pin 1601 is coupled(e.g., bolted) to the bottom of the antenna structure, which isconducting ground plane 1602.

Separate from conducting ground plane 1602 is interstitial conductor1603, which is an internal conductor. In one embodiment, conductingground plane 1602 and interstitial conductor 1603 are parallel to eachother. In one embodiment, the distance between ground plane 1602 andinterstitial conductor 1603 is 0.1-0.15″. In another embodiment, thisdistance may be λ/2, where λ, is the wavelength of the travelling waveat the frequency of operation.

Ground plane 1602 is separated from interstitial conductor 1603 via aspacer 1604. In one embodiment, spacer 1604 is a foam or air-likespacer. In one embodiment, spacer 1604 comprises a plastic spacer.

On top of interstitial conductor 1603 is dielectric layer 1605. In oneembodiment, dielectric layer 1605 is plastic. The purpose of dielectriclayer 1605 is to slow the travelling wave relative to free spacevelocity. In one embodiment, dielectric layer 1605 slows the travellingwave by 30% relative to free space. In one embodiment, the range ofindices of refraction that are suitable for beam forming are 1.2-1.8,where free space has by definition an index of refraction equal to 1.Other dielectric spacer materials, such as, for example, plastic, may beused to achieve this effect. Note that materials other than plastic maybe used as long as they achieve the desired wave slowing effect.Alternatively, a material with distributed structures may be used asdielectric 1605, such as periodic sub-wavelength metallic structuresthat can be machined or lithographically defined, for example.

An RF-array 1606 is on top of dielectric 1605. In one embodiment, thedistance between interstitial conductor 1603 and RF-array 1606 is0.1-0.15″. In another embodiment, this distance may be λ_(eff)/2, whereλ_(eff) is the effective wavelength in the medium at the designfrequency.

The antenna includes sides 1607 and 1608. Sides 1607 and 1608 are angledto cause a travelling wave feed from coax pin 1601 to be propagated fromthe area below interstitial conductor 1603 (the spacer layer) to thearea above interstitial conductor 1603 (the dielectric layer) viareflection. In one embodiment, the angle of sides 1607 and 1608 are at45° angles. In an alternative embodiment, sides 1607 and 1608 could bereplaced with a continuous radius to achieve the reflection. While FIG.27 shows angled sides that have angle of 45 degrees, other angles thataccomplish signal transmission from lower level feed to upper level feedmay be used. That is, given that the effective wavelength in the lowerfeed will generally be different than in the upper feed, some deviationfrom the ideal 45° angles could be used to aid transmission from thelower to the upper feed level. For example, in another embodiment, the45° angles are replaced with a single step. The steps on one end of theantenna go around the dielectric layer, interstitial the conductor, andthe spacer layer. The same two steps are at the other ends of theselayers.

In operation, when a feed wave is fed in from coaxial pin 1601, the wavetravels outward concentrically oriented from coaxial pin 1601 in thearea between ground plane 1602 and interstitial conductor 1603. Theconcentrically outgoing waves are reflected by sides 1607 and 1608 andtravel inwardly in the area between interstitial conductor 1603 and RFarray 1606. The reflection from the edge of the circular perimetercauses the wave to remain in phase (i.e., it is an in-phase reflection).The travelling wave is slowed by dielectric layer 1605. At this point,the travelling wave starts interacting and exciting with elements in RFarray 1606 to obtain the desired scattering.

To terminate the travelling wave, a termination 1609 is included in theantenna at the geometric center of the antenna. In one embodiment,termination 1609 comprises a pin termination (e.g., a 50 Ω pin). Inanother embodiment, termination 1609 comprises an RF absorber thatterminates unused energy to prevent reflections of that unused energyback through the feed structure of the antenna. These could be used atthe top of RF array 1606.

FIG. 28 illustrates another embodiment of the antenna system with anoutgoing wave. Referring to FIG. 28, two ground planes 1610 and 1611 aresubstantially parallel to each other with a dielectric layer 1612 (e.g.,a plastic layer, etc.) in between ground planes. RF absorbers 1619(e.g., resistors) couple the two ground planes 1610 and 1611 together. Acoaxial pin 1615 (e.g., 50 Ω) feeds the antenna. An RF array 1616 is ontop of dielectric layer 1612 and ground plane 1611.

In operation, a feed wave is fed through coaxial pin 1615 and travelsconcentrically outward and interacts with the elements of RF array 1616.

The cylindrical feed in both the antennas of FIGS. 25 and 26 improvesthe service angle of the antenna. Instead of a service angle of plus orminus forty five degrees azimuth (±45° Az) and plus or minus twenty fivedegrees elevation (±25° El), in one embodiment, the antenna system has aservice angle of seventy five degrees (75°) from the bore sight in alldirections. As with any beam forming antenna comprised of manyindividual radiators, the overall antenna gain is dependent on the gainof the constituent elements, which themselves are angle-dependent. Whenusing common radiating elements, the overall antenna gain typicallydecreases as the beam is pointed further off bore sight. At 75 degreesoff bore sight, significant gain degradation of about 6 dB is expected.

Embodiments of the antenna having a cylindrical feed solve one or moreproblems. These include dramatically simplifying the feed structurecompared to antennas fed with a corporate divider network and thereforereducing total required antenna and antenna feed volume; decreasingsensitivity to manufacturing and control errors by maintaining high beamperformance with coarser controls (extending all the way to simplebinary control); giving a more advantageous side lobe pattern comparedto rectilinear feeds because the cylindrically oriented feed wavesresult in spatially diverse side lobes in the far field; and allowingpolarization to be dynamic, including allowing left-hand circular,right-hand circular, and linear polarizations, while not requiring apolarizer.

Array of Wave Scattering Elements

RF array 1606 of FIG. 27 and RF array 1616 of FIG. 28 include a wavescattering subsystem that includes a group of patch antennas (i.e.,scatterers) that act as radiators. This group of patch antennascomprises an array of scattering metamaterial elements.

In one embodiment, each scattering element in the antenna system is partof a unit cell that consists of a lower conductor, a dielectricsubstrate and an upper conductor that embeds a complementary electricinductive-capacitive resonator (“complementary electric LC” or “CELC”)that is etched in or deposited onto the upper conductor.

In one embodiment, a liquid crystal (LC) is injected in the gap aroundthe scattering element. Liquid crystal is encapsulated in each unit celland separates the lower conductor associated with a slot from an upperconductor associated with its patch. Liquid crystal has a permittivitythat is a function of the orientation of the molecules comprising theliquid crystal, and the orientation of the molecules (and thus thepermittivity) can be controlled by adjusting the bias voltage across theliquid crystal. Using this property, the liquid crystal acts as anon/off switch for the transmission of energy from the guided wave to theCELC. When switched on, the CELC emits an electromagnetic wave like anelectrically small dipole antenna.

Controlling the thickness of the LC increases the beam switching speed.A fifty percent (50%) reduction in the gap between the lower and theupper conductor (the thickness of the liquid crystal) results in afourfold increase in speed. In another embodiment, the thickness of theliquid crystal results in a beam switching speed of approximatelyfourteen milliseconds (14 ms). In one embodiment, the LC is doped in amanner well-known in the art to improve responsiveness so that a sevenmillisecond (7 ms) requirement can be met.

The CELC element is responsive to a magnetic field that is appliedparallel to the plane of the CELC element and perpendicular to the CELCgap complement. When a voltage is applied to the liquid crystal in themetamaterial scattering unit cell, the magnetic field component of theguided wave induces a magnetic excitation of the CELC, which, in turn,produces an electromagnetic wave in the same frequency as the guidedwave.

The phase of the electromagnetic wave generated by a single CELC can beselected by the position of the CELC on the vector of the guided wave.Each cell generates a wave in phase with the guided wave parallel to theCELC. Because the CELCs are smaller than the wave length, the outputwave has the same phase as the phase of the guided wave as it passesbeneath the CELC.

In one embodiment, the cylindrical feed geometry of this antenna systemallows the CELC elements to be positioned at forty-five-degree (45°)angles to the vector of the wave in the wave feed. This position of theelements enables control of the polarization of the free space wavegenerated from or received by the elements. In one embodiment, the CELCsare arranged with an inter-element spacing that is less than afree-space wavelength of the operating frequency of the antenna. Forexample, if there are four scattering elements per wavelength, theelements in the 30 GHz transmit antenna will be approximately 2.5 mm(i.e., ¼th the 10 mm free-space wavelength of 30 GHz).

In one embodiment, the CELCs are implemented with patch antennas thatinclude a patch co-located over a slot with liquid crystal between thetwo. In this respect, the metamaterial antenna acts like a slotted(scattering) wave guide. With a slotted wave guide, the phase of theoutput wave depends on the location of the slot in relation to theguided wave.

Cell Placement

In one embodiment, the antenna elements are placed on the cylindricalfeed antenna aperture in a way that allows for a systematic matrix drivecircuit. The placement of the cells includes placement of thetransistors for the matrix drive. FIG. 29 illustrates one embodiment ofthe placement of matrix drive circuitry with respect to antennaelements.

Referring to FIG. 29, row controller 1701 is coupled to transistors 1711and 1712, via row select signals Row1 and Row2, respectively, and columncontroller 1702 is coupled to transistors 1711 and 1712 via columnselect signal Column1. Transistor 1711 is also coupled to antennaelement 1721 via connection to patch 1731, while transistor 1712 iscoupled to antenna element 1722 via connection to patch 1732.

In an initial approach to realize matrix drive circuitry on thecylindrical feed antenna with unit cells placed in a non-regular grid,two steps are performed. In the first step, the cells are placed onconcentric rings and each of the cells is connected to a transistor thatis placed beside the cell and acts as a switch to drive each cellseparately. In the second step, the matrix drive circuitry is built inorder to connect every transistor with a unique address as the matrixdrive approach requires. Because the matrix drive circuit is built byrow and column traces (similar to LCDs) but the cells are placed onrings, there is no systematic way to assign a unique address to eachtransistor. This mapping problem results in very complex circuitry tocover all the transistors and leads to a significant increase in thenumber of physical traces to accomplish the routing. Because of the highdensity of cells, those traces disturb the RF performance of the antennadue to coupling effect. Also, due to the complexity of traces and highpacking density, the routing of the traces cannot be accomplished bycommercially available layout tools.

In one embodiment, the matrix drive circuitry is predefined before thecells and transistors are placed. This ensures a minimum number oftraces that are necessary to drive all the cells, each with a uniqueaddress. This strategy reduces the complexity of the drive circuitry andsimplifies the routing, which subsequently improves the RF performanceof the antenna.

More specifically, in one approach, in the first step, the cells areplaced on a regular rectangular grid composed of rows and columns thatdescribe the unique address of each cell. In the second step, the cellsare grouped and transformed to concentric circles while maintainingtheir address and connection to the rows and columns as defined in thefirst step. A goal of this transformation is not only to put the cellson rings but also to keep the distance between cells and the distancebetween rings constant over the entire aperture. In order to accomplishthis goal, there are several ways to group the cells.

In one embodiment, a TFT package is used to enable placement and uniqueaddressing in the matrix drive. FIG. 30 illustrates one embodiment of aTFT package. Referring to FIG. 30, a TFT and a hold capacitor 1803 isshown with input and output ports. There are two input ports connectedto traces 1801 and two output ports connected to traces 1802 to connectthe TFTs together using the rows and columns. In one embodiment, the rowand column traces cross in 90° angles to reduce, and potentiallyminimize, the coupling between the row and column traces. In oneembodiment, the row and column traces are on different layers.

An Example System Embodiment

In one embodiment, the combined antenna apertures are used in atelevision system that operates in conjunction with a set top box. Forexample, in the case of a dual reception antenna, satellite signalsreceived by the antenna are provided to a set top box (e.g., a DirecTVreceiver) of a television system. More specifically, the combinedantenna operation is able to simultaneously receive RF signals at twodifferent frequencies and/or polarizations. That is, one sub-array ofelements is controlled to receive RF signals at one frequency and/orpolarization, while another sub-array is controlled to receive signalsat another, different frequency and/or polarization. These differencesin frequency or polarization represent different channels being receivedby the television system. Similarly, the two antenna arrays can becontrolled for two different beam positions to receive channels from twodifferent locations (e.g., two different satellites) to simultaneouslyreceive multiple channels.

FIG. 31 is a block diagram of one embodiment of a communication systemthat performs dual reception simultaneously in a television system.Referring to FIG. 31, antenna 1401 includes two spatially interleavedantenna apertures operable independently to perform dual receptionsimultaneously at different frequencies and/or polarizations asdescribed above. Note that while only two spatially interleaved antennaoperations are mentioned, the TV system may have more than two antennaapertures (e.g., 3, 4, 5, etc. antenna apertures).

In one embodiment, antenna 1401, including its two interleaved slottedarrays, is coupled to diplexer 1430. The coupling may include one ormore feeding networks that receive the signals from elements of the twoslotted arrays to produce two signals that are fed into diplexer 1430.In one embodiment, diplexer 1430 is a commercially available diplexer(e.g., model PB1081WA Ku-band sitcom diplexer from A1 Microwave).

Diplexer 1430 is coupled to a pair of low noise block down converters(LNBs) 1426 and 1427, which perform a noise filtering function, a downconversion function, and amplification in a manner well-known in theart. In one embodiment, LNBs 1426 and 1427 are in an out-door unit(ODU). In another embodiment, LNBs 1426 and 1427 are integrated into theantenna apparatus. LNBs 1426 and 1427 are coupled to a set top box 1402,which is coupled to television 1403.

Set top box 1402 includes a pair of analog-to-digital converters (ADCs)1421 and 1422, which are coupled to LNBs 1426 and 1427, to convert thetwo signals output from diplexer 1430 into digital format.

Once converted to digital format, the signals are demodulated bydemodulator 1423 and decoded by decoder 1424 to obtain the encoded dataon the received waves. The decoded data is then sent to controller 1425,which sends it to television 1403.

Controller 1450 controls antenna 1401, including the interleaved slottedarray elements of both antenna apertures on the single combined physicalaperture.

An Example of a Full Duplex Communication System

In another embodiment, the combined antenna apertures are used in a fullduplex communication system. FIG. 32 is a block diagram of anotherembodiment of a communication system having simultaneous transmit andreceive paths. While only one transmit path and one receive path areshown, the communication system may include more than one transmit pathand/or more than one receive path.

Referring to FIG. 32, antenna 1401 includes two spatially interleavedantenna arrays operable independently to transmit and receivesimultaneously at different frequencies as described above. In oneembodiment, antenna 1401 is coupled to diplexer 1445. The coupling maybe by one or more feeding networks. In one embodiment, in the case of aradial feed antenna, diplexer 1445 combines the two signals and theconnection between antenna 1401 and diplexer 1445 is a single broad-bandfeeding network that can carry both frequencies.

Diplexer 1445 is coupled to a low noise block down converter (LNBs)1427, which performs a noise filtering function and a down conversionand amplification function in a manner well-known in the art. In oneembodiment, LNB 1427 is in an out-door unit (ODU). In anotherembodiment, LNB 1427 is integrated into the antenna apparatus. LNB 1427is coupled to a modem 1460, which is coupled to computing system 1440(e.g., a computer system, modem, etc.).

Modem 1460 includes an analog-to-digital converter (ADC) 1422, which iscoupled to LNB 1427, to convert the received signal output from diplexer1445 into digital format. Once converted to digital format, the signalis demodulated by demodulator 1423 and decoded by decoder 1424 to obtainthe encoded data on the received wave. The decoded data is then sent tocontroller 1425, which sends it to computing system 1440.

Modem 1460 also includes an encoder 1430 that encodes data to betransmitted from computing system 1440. The encoded data is modulated bymodulator 1431 and then converted to analog by digital-to-analogconverter (DAC) 1432. The analog signal is then filtered by a BUC(up-convert and high pass amplifier) 1433 and provided to one port ofdiplexer 1445. In one embodiment, BUC 1433 is in an out-door unit (ODU).

Diplexer 1445 operating in a manner well-known in the art provides thetransmit signal to antenna 1401 for transmission.

Controller 1450 controls antenna 1401, including the two arrays ofantenna elements on the single combined physical aperture.

Note that the full duplex communication system shown in FIG. 32 has anumber of applications, including but not limited to, internetcommunication, vehicle communication (including software updating), etc.

There is a number of example embodiments described herein.

Example 1 is a method for controlling an antenna having a plurality ofradio frequency (RF) radiating antenna elements, the method comprising:performing modulation for a frequency range, including determining afirst set of modulation values for achievable modulation statesassociated with the frequency range; mapping modulation valuesassociated with the achievable modulation states to one or more controlparameters; and controlling the radiating antenna elements using the oneor more control parameters to perform beam forming.

Example 2 is the method of example 1 that may optionally include thatperforming modulation for a frequency range occurs in response todetermining instantaneous bandwidth (IBW) requirements would not be metby a beam formed by the antenna with a second set of modulation values.

Example 3 is the method of example 3 that may optionally include thatdetermining instantaneous bandwidth (IBW) requirements are not met by abeam formed by the antenna with a second set of modulation values isperformed after determining gain is sufficient to allow for adjustmentsto the frequency range associated with the second set of modulationvalues in order to satisfy the IBW requirements.

Example 4 is the method of example 1 that may optionally include mappinga desired modulation to the achievable modulation states.

Example 5 is the method of example 4 that may optionally include thatmapping the desired modulation to the achievable modulation states isbased on Euclidian distance between required and achievablepolarizabilities and each of the achievable modulation statescorresponds to voltages applied to the antenna elements of the antennato induce magnetic dipole moments with achievable polarizabilities.

Example 6 is the method of example 4 that may optionally include thatthe RF radiating antenna elements comprise tunable elements in ametasurface and mapping the desired modulation to the achievablemodulation states comprises approximating a set of requiredpolarizabilities with a set of the tunable elements in the metasurface.

Example 7 is the method of example 1 that may optionally include thatthe one or more control parameters comprise a voltage to be applied toeach of the RF radiating antenna elements.

Example 8 is the method of example 1 that may optionally includeobtaining the desired modulation, wherein the desired modulation isbased on one or more of: location of at least a subset of the RFradiating antenna elements, beam pointing direction and polarization,and wave propagation in a feed of the antenna.

Example 9 is an antenna comprising: an antenna aperture having aplurality of RF radiating antenna elements; a controller coupled to theantenna aperture and having modulation logic to perform modulation for afrequency range, the modulation logic operable to determine a first setof modulation values for achievable modulation states associated withthe frequency range and map modulation values associated with theachievable modulation states to one or more control parameters; anddrive circuitry coupled to the antenna aperture and the controller tocontrol the RF radiating antenna elements using the one or more controlparameters to perform beam forming.

Example 10 is the antenna of example 9 that may optionally include thatthe modulation logic is operable to perform modulation for a frequencyrange occurs in response to determining instantaneous bandwidth (IBW)requirements are not met by a beam formed by the antenna with a secondset of modulation values.

Example 11 is the antenna of example 10 that may optionally include thatthe modulation logic is operable to determine instantaneous bandwidth(IBW) requirements would not be met by a beam formed by the antenna witha second set of modulation values after determining gain is sufficientto allow for adjustments to the frequency range associated with thesecond set of modulation values in order to satisfy the IBWrequirements.

Example 12 is the antenna of example 9 that may optionally include thatthe modulation logic is further operable to map a desired modulation tothe achievable modulation states.

Example 13 is the antenna of example 12 that may optionally include thatthe modulation logic maps the desired modulation to the achievablemodulation states based on Euclidian distance between required andachievable polarizabilities and each of the achievable modulation statescorresponds to voltages applied to the antenna elements of the antennato induce magnetic dipole moments with achievable polarizabilities.

Example 14 is the antenna of example 12 that may optionally include thatthe RF radiating antenna elements comprise tunable elements in ametasurface and the modulation logic maps the desired modulation to theachievable modulation states by approximating a set of requiredpolarizabilities with a set of the tunable elements in the metasurface.

Example 15 is the antenna of example 9 that may optionally include thatthe one or more control parameters comprise a voltage to be applied toeach of the RF radiating antenna elements.

Example 16 is the antenna of example 9 that may optionally include thatthe modulation logic is further operable to obtain the desiredmodulation, wherein the desired modulation is based on one or more of:location of at least a subset of the RF radiating antenna elements, beampointing direction and polarization, and wave propagation in a feed ofthe antenna.

Example 17 is one or more non-transitory computer readable storage mediahaving instructions stored thereupon which, when executed by an antennahaving at least a processor and a memory therein, cause antenna toperform operations comprising: performing modulation for a frequencyrange, including determining a first set of modulation values forachievable modulation states associated with the frequency range;mapping modulation values associated with the achievable modulationstates to one or more control parameters; and controlling the radiatingantenna elements using the one or more control parameters to performbeam forming.

Example 18 is the one or more non-transitory computer readable storagemedia of example 17 that may optionally include that performingmodulation for a frequency range occurs in response to determininginstantaneous bandwidth (IBW) requirements would not be met by a beamformed by the antenna with a second set of modulation values.

Example 19 is the one or more non-transitory computer readable storagemedia of example 18 that may optionally include that determininginstantaneous bandwidth (IBW) requirements would not be met by a beamformed by the antenna with a second set of modulation values isperformed after determining gain is sufficient to allow for adjustmentsto the frequency range associated with the second set of modulationvalues in order to satisfy the IBW requirements.

Example 20 is the one or more non-transitory computer readable storagemedia of example 17 that may optionally include that mapping a desiredmodulation to the achievable modulation states.

Example 21 is the one or more non-transitory computer readable storagemedia of example 20 that may optionally include that mapping the desiredmodulation to the achievable modulation states is based on Euclidiandistance between required and achievable polarizabilities and each ofthe achievable modulation states corresponds to voltages applied to theantenna elements of the antenna to induce magnetic dipole moments withachievable polarizabilities.

Example 22 is a method comprising: a) determining, for a first frequencywindow, a desired modulation associated with a desired impedance foreach radio frequency (RF) radiating antenna element of a plurality of RFradiating antenna elements of an antenna, using a holographic equationthat relates a feed wave to a desired radiation pattern; b) mapping thedesired modulation to achievable modulation states using Euclideanmapping; c) determining whether gain associated with a modulationpattern is above a threshold; d) determining whether requirements forinstantaneous bandwidth (IBW) would be met previously by the firstfrequency window; e) adjusting the first frequency window by increasingsize of the first frequency window to create a second frequency windowif the gain is above the threshold and requirements for IBW would not bemet by the first window; and f) repeating a)-c) using the secondfrequency window instead of the first frequency window.

Example 23 is the method of example 22 that may optionally include thatthe first frequency window comprises a single frequency.

Example 24 is the method of example 22 that may optionally include g)determining whether requirements for instantaneous bandwidth (IBW) havebeen met for the modulation with the second frequency window; and h) ifrequirements for IBW are met, then mapping modulation values associatedwith the achievable modulation states to one or more control parameters,and controlling radio frequency (RF) radiating antenna elements usingthe one or more control parameters to perform beam forming.

Example 25 is the method of example 24 that may optionally include, ifrequirements for IBW are not met but the gain is above the threshold,then h) adjusting the second frequency window by increasing size of thesecond frequency window to create a third frequency window; and i)repeating a)-c) using the third frequency window instead of the secondfrequency window.

Example 26 is the method of example 24 that may optionally include thatmapping a desired modulation to achievable modulation states is based onEuclidian distance.

Example 27 is the method of example 24 that may optionally include thatthe one or more control parameters comprise a voltage to be applied toeach of the RF radiating antenna elements.

Example 28 is the method of example 27 that may optionally include thatthe RF radiating antenna elements comprise tunable elements in ametasurface and mapping the desired modulation to achievable modulationstates comprises approximating a set of required polarizabilities with aset of the tunable elements in the metasurface.

Some portions of the detailed descriptions above are presented in termsof algorithms and symbolic representations of operations on data bitswithin a computer memory. These algorithmic descriptions andrepresentations are the means used by those skilled in the dataprocessing arts to most effectively convey the substance of their workto others skilled in the art. An algorithm is here, and generally,conceived to be a self-consistent sequence of steps leading to a desiredresult. The steps are those requiring physical manipulations of physicalquantities. Usually, though not necessarily, these quantities take theform of electrical or magnetic signals capable of being stored,transferred, combined, compared, and otherwise manipulated. It hasproven convenient at times, principally for reasons of common usage, torefer to these signals as bits, values, elements, symbols, characters,terms, numbers, or the like.

It should be borne in mind, however, that all of these and similar termsare to be associated with the appropriate physical quantities and aremerely convenient labels applied to these quantities. Unlessspecifically stated otherwise as apparent from the following discussion,it is appreciated that throughout the description, discussions utilizingterms such as “processing” or “computing” or “calculating” or“determining” or “displaying” or the like, refer to the action andprocesses of a computer system, or similar electronic computing device,that manipulates and transforms data represented as physical(electronic) quantities within the computer system's registers andmemories into other data similarly represented as physical quantitieswithin the computer system memories or registers or other suchinformation storage, transmission or display devices.

The present invention also relates to apparatus for performing theoperations herein. This apparatus may be specially constructed for therequired purposes, or it may comprise a general-purpose computerselectively activated or reconfigured by a computer program stored inthe computer. Such a computer program may be stored in a computerreadable storage medium, such as, but is not limited to, any type ofdisk including floppy disks, optical disks, CD-ROMs, andmagnetic-optical disks, read-only memories (ROMs), random accessmemories (RAMs), EPROMs, EEPROMs, magnetic or optical cards, or any typeof media suitable for storing electronic instructions, and each coupledto a computer system bus.

The algorithms and displays presented herein are not inherently relatedto any particular computer or other apparatus. Various general-purposesystems may be used with programs in accordance with the teachingsherein, or it may prove convenient to construct more specializedapparatus to perform the required method steps. The required structurefor a variety of these systems will appear from the description below.In addition, the present invention is not described with reference toany particular programming language. It will be appreciated that avariety of programming languages may be used to implement the teachingsof the invention as described herein.

A machine-readable medium includes any mechanism for storing ortransmitting information in a form readable by a machine (e.g., acomputer). For example, a machine-readable medium includes read onlymemory (“ROM”); random access memory (“RAM”); magnetic disk storagemedia; optical storage media; flash memory devices; etc.

Whereas many alterations and modifications of the present invention willno doubt become apparent to a person of ordinary skill in the art afterhaving read the foregoing description, it is to be understood that anyparticular embodiment shown and described by way of illustration is inno way intended to be considered limiting. Therefore, references todetails of various embodiments are not intended to limit the scope ofthe claims which in themselves recite only those features regarded asessential to the invention.

We claim:
 1. A method for controlling an antenna having a plurality of radio frequency (RF) radiating antenna elements, the method comprising: performing modulation for a frequency range, including determining a first set of modulation values for achievable modulation states associated with the frequency range; mapping modulation values associated with the achievable modulation states to one or more control parameters; and controlling the radiating antenna elements using the one or more control parameters to perform beam forming.
 2. The method of claim 1 wherein performing modulation for a frequency range occurs in response to determining instantaneous bandwidth (IBW) requirements would not be met by a beam formed by the antenna with a second set of modulation values.
 3. The method of claim 2 wherein determining instantaneous bandwidth (IBW) requirements are not met by a beam formed by the antenna with a second set of modulation values is performed after determining gain is sufficient to allow for adjustments to the frequency range associated with the second set of modulation values in order to satisfy the IBW requirements.
 4. The method of claim 1 further comprising mapping a desired modulation to the achievable modulation states.
 5. The method of claim 4 wherein mapping the desired modulation to the achievable modulation states is based on Euclidian distance between required and achievable polarizabilities and each of the achievable modulation states corresponds to voltages applied to the antenna elements of the antenna to induce magnetic dipole moments with achievable polarizabilities.
 6. The method of claim 4 wherein the RF radiating antenna elements comprise tunable elements in a metasurface and mapping the desired modulation to the achievable modulation states comprises approximating a set of required polarizabilities with a set of the tunable elements in the metasurface.
 7. The method defined in claim 1 wherein the one or more control parameters comprise a voltage to be applied to each of the RF radiating antenna elements.
 8. The method defined in claim 1 further comprising obtaining the desired modulation, wherein the desired modulation is based on one or more of: location of at least a subset of the RF radiating antenna elements, beam pointing direction and polarization, and wave propagation in a feed of the antenna.
 9. An antenna comprising: an antenna aperture having a plurality of RF radiating antenna elements; a controller coupled to the antenna aperture and having modulation logic to perform modulation for a frequency range, the modulation logic operable to determine a first set of modulation values for achievable modulation states associated with the frequency range and map modulation values associated with the achievable modulation states to one or more control parameters; and drive circuitry coupled to the antenna aperture and the controller to control the RF radiating antenna elements using the one or more control parameters to perform beam forming.
 10. The antenna of claim 9 wherein the modulation logic is operable to perform modulation for a frequency range occurs in response to determining instantaneous bandwidth (IBW) requirements are not met by a beam formed by the antenna with a second set of modulation values.
 11. The antenna of claim 10 wherein the modulation logic is operable to determine instantaneous bandwidth (IBW) requirements would not be met by a beam formed by the antenna with a second set of modulation values after determining gain is sufficient to allow for adjustments to the frequency range associated with the second set of modulation values in order to satisfy the IBW requirements.
 12. The antenna of claim 9 wherein the modulation logic is further operable to map a desired modulation to the achievable modulation states.
 13. The antenna of claim 12 wherein the modulation logic maps the desired modulation to the achievable modulation states based on Euclidian distance between required and achievable polarizabilities and each of the achievable modulation states corresponds to voltages applied to the antenna elements of the antenna to induce magnetic dipole moments with achievable polarizabilities.
 14. The antenna of claim 12 wherein the RF radiating antenna elements comprise tunable elements in a metasurface and the modulation logic maps the desired modulation to the achievable modulation states by approximating a set of required polarizabilities with a set of the tunable elements in the metasurface.
 15. The antenna defined in claim 9 wherein the one or more control parameters comprise a voltage to be applied to each of the RF radiating antenna elements.
 16. The antenna defined in claim 9 wherein the modulation logic is further operable to obtain the desired modulation, wherein the desired modulation is based on one or more of: location of at least a subset of the RF radiating antenna elements, beam pointing direction and polarization, and wave propagation in a feed of the antenna.
 17. One or more non-transitory computer readable storage media having instructions stored thereupon which, when executed by an antenna having at least a processor and a memory therein, cause antenna to perform operations comprising: performing modulation for a frequency range, including determining a first set of modulation values for achievable modulation states associated with the frequency range; mapping modulation values associated with the achievable modulation states to one or more control parameters; and controlling the radiating antenna elements using the one or more control parameters to perform beam forming.
 18. The one or more non-transitory computer readable storage media of claim 17 wherein performing modulation for a frequency range occurs in response to determining instantaneous bandwidth (IBW) requirements would not be met by a beam formed by the antenna with a second set of modulation values.
 19. The one or more non-transitory computer readable storage media of claim 18 wherein determining instantaneous bandwidth (IBW) requirements would not be met by a beam formed by the antenna with a second set of modulation values is performed after determining gain is sufficient to allow for adjustments to the frequency range associated with the second set of modulation values in order to satisfy the IBW requirements.
 20. The one or more non-transitory computer readable storage media of claim 17 further comprising mapping a desired modulation to the achievable modulation states.
 21. The one or more non-transitory computer readable storage media of claim 20 wherein mapping the desired modulation to the achievable modulation states is based on Euclidian distance between required and achievable polarizabilities and each of the achievable modulation states corresponds to voltages applied to the antenna elements of the antenna to induce magnetic dipole moments with achievable polarizabilities.
 22. A method comprising: a) determining, for a first frequency window, a desired modulation associated with a desired impedance for each radio frequency (RF) radiating antenna element of a plurality of RF radiating antenna elements of an antenna, using a holographic equation that relates a feed wave to a desired radiation pattern; b) mapping the desired modulation to achievable modulation states using Euclidean mapping; c) determining whether gain associated with a modulation pattern is above a threshold; d) determining whether requirements for instantaneous bandwidth (113W) would be met previously by the first frequency window; e) adjusting the first frequency window by increasing size of the first frequency window to create a second frequency window if the gain is above the threshold and requirements for IBW would not be met by the first window; and f) repeating a)-c) using the second frequency window instead of the first frequency window.
 23. The method of claim 22 wherein the first frequency window comprises a single frequency.
 24. The method of claim 22 further comprising: g) determining whether requirements for instantaneous bandwidth (IBW) have been met for the modulation with the second frequency window; and h) if requirements for IBW are met, then mapping modulation values associated with the achievable modulation states to one or more control parameters, and controlling radio frequency (RF) radiating antenna elements using the one or more control parameters to perform beam forming.
 25. The method of claim 24 further comprising: if requirements for IBW are not met but the gain is above the threshold, then h) adjusting the second frequency window by increasing size of the second frequency window to create a third frequency window; and i) repeating a)-c) using the third frequency window instead of the second frequency window.
 26. The method of claim 24 wherein mapping a desired modulation to achievable modulation states is based on Euclidian distance.
 27. The method of claim 24 wherein the one or more control parameters comprise a voltage to be applied to each of the RF radiating antenna elements.
 28. The method defined in claim 27 wherein the RF radiating antenna elements comprise tunable elements in a metasurface and mapping the desired modulation to achievable modulation states comprises approximating a set of required polarizabilities with a set of the tunable elements in the metasurface. 